Renting a van from Norway to UK

I very much doubt you will be able to do this as a one way rent. Someone would have to drive the car (Norwegian registered) all the way back to Norway. And you would have to pay for this (flight to the UK, accommodation, time, fuel etc.). Even a one way rental from Bergen to Brussels will have a horrible surcharge.
Some car hire companies ban (or surcharge you) for taking cars on ferries, and/or crossing the Channel. They seem to think having a steering wheel on the wrong side is an increased accident risk.

Brussels to London is easy, Eurostar High Speed train, it only takes 2 hours. Buy up to 6 months ahead of time for cheap fares: https://www.eurostar.com

A normal licence is valid for vehicles up to 8 seats. 7 + 2 aunts = 9 and requires a special licence.
What you call a van is significantly larger (height and width) than a normal car. Most car parks (open air and indoor) have a height limit of 1.85 metres, vans are higher than this, so you will have trouble with parking, including hotel car parks.

How much time for this mega trip?
You also have to decide what places you are visiting. Cars (and vans) are OK for outside cities. In cities they are a liability; not permitted, or restricted, and parking is not available at most tourist destinations in cities, you have to park some distance away and take public transport.

Interesting. I'll look into rail.
In the US, we could rent and drive this distance one way with no problem but of course all within one country. We drive a full size van as our everyday auto. I appreciate learning about how to accomplish this trip in unfamiliar territory! Thanks!

One way rental within a country is no problem. Taking a vehicle into other countries is usually OK, though into or out of the UK is considered an extra risk as the steering wheel is on the wrong side. Renting out foreign-registered vehicles owned by foreign companies is usually not allowed, so it has to be driven back, at the renters expense. I imagine if a company in your town rented out Mexican or Canadian registered vehicles they would also be breaking some laws.
The common solution is to plan a circular tour.

I think you need to get a rough plan of what you want to see: cities, areas, how long in each. Then fit the transport to the destinations.

How long is a good question. You need 2+ nights at each location. Big cities (London, Paris etc.) need 4-5 nights or more.
For planning road routes I recommend https://www.viamichelin.com/
This will give distances, times, plus cost (fuel + tolls). Time estimates assume you have no stops. Add 25%.

If you think of using train for some legs, reads this primer (followed by the rest of the site): https://www.seat61.com/Europe-train-travel.htm
Trains on High Speed routes travel at nearly 200mph city centre to city centre, well above your van's top speed!

And, if you are planning a really long trip, don't forget non EU/Schengen citizens are limited to 90 days in any 180 in the Schengen Area.

If you put 9 people in a 9 seater mini-bus where will you put the luggage? In Europe, you generally get space behind the second row of seats which could be used for luggage, but if you put the third row of seats up you usually have them right to the back window and no room (or extremely small room) behind.

What brand vehicle have you looked at?

Norway is one of the most expensive countries in Europe, so you may want to consider starting elsewhere in Europe that would offer better value. A one way car repatriation will cost a fortune.

Most European people carriers (a van is a commercial vehicle with no windows in Europe) carry a maximum of 7 people. Many European hire companies will not permit you to take their vehicles to the UK. As has already been suggested, take the Eurostar.

Ideally, you want a right hand drive vehicle in the UK.

Great links and information! Thanks!
I'm already switching gears and looking at rail travel.

Our trip will consist of flying from Chicago to Bergen Norway (possibly laying over in Reykjavik for 2 nights to break up the flight time and get over the jet lag.)
Our main objectives are to see fjords and experience Norway for 5 days (from Bergen to Flam to Oslo) and also to spend 4 days in and around London before taking a transatlantic cruise back to the US that leaves from Dover.
Our desire to see more countries has us wanting to travel via road (now probably rail) from Norway to UK. We'd have between 5-7 days (if we drop layover in Reykjavik or leave Chicago sooner.) I know that's hectic, but it would be nice to see the scenery and have even a few hours to experience the cities we stop and stay in.
My rough plan (so far) would be to leave from Oslo and stop in Copenhagen for 2 nights (going to Tivoli Gardens with the kids one day), Hamburg for 1 night (but I've already read to try to visit a smaller city rather than a large one), (Consider going to Amsterdam if we make more time), Brussels for one night, then get to London for 4 nights before embarking on the cruise to the US.

This pre-cruise trip will be slightly longer than 2 weeks long (16 nights.)

The Scandinavian countries are deadly expensive in every way. There is no way such a trip would be affordable. A burger, fries and Coke at TGI Fridays is $30 per person, for example. We find the best way to see the region affordably is by cruise ships.
Travel is logistically difficult for 5 people. Go up to 7 and you will find full-size vans hard to find and numerous hotel rooms are required. Jump to nine people, and you are chartering a small tour bus with a driver. That is when a good travel agent is needed.
We were in Ireland a week ago, and diesel fuel was $6.04 per U.S. gallon. Fuel would be higher in the Scandinavian countries even. That is enough to make you consider travel by bus or train.
Many people do best to find a couple of centrally located cities, rent a house and do day trips from there. The days of my taking in 5-6 countries on a single trip is over as traveling slower is best.

If seven to nine people are going on a cruise after two weeks in Europe, then money is not the primary consideration. But you mention cost as a driver for your decision-making. Thank you for mentioning that it is your second, not first, trip to Europe. If so, it puzzles me that you are considering so much driving for the two-week, land-part of your vacation. (Because of overseas travel, and the need to be near the cruise Embarkation the night before, the usual warnings here about having only 12, not fourteen days available, apply - even though you're not going home at the end of the land trip.) I don't believe you have enough days for the number of visits you want to make.

I don't have a big family, but I would think that you have to add to the usual delays of getting off the highway and into a destination city and parked, the need to feed and bathroom and organize such a large number of people. I worry about organizing multiple train tickets for 9 people if you drop the hope of driving.

You know exactly how much luggage space an American ten-passenger van has in it. Do you really think you can accommodate your own luggage? I guess you mentioned "home-schooled" to indicate how close your family is. But you still need seven or nine clean tee shirts every morning! BTW, have you strategized laundry? I believe that vehicles on international ferries are quite expensive - but your problem may be getting a reservation at a day and time you prefer. I suppose they still use their biggest boats in September. Have you read past replies on this board about rough water in the English Channel? Are you planning to drive and return a left-hand drive vehicle in England? Your plan has too much of an air of "Southwestern American national parks" or "Sacramento to Los Angeles" by American Interstate highway. Did you rent a car during the first trip to Europe? How did that work out? It can't be taken for granted that European hotels have free (or even, any on-premises) parking, especially in big cities.

Although Bergen is not an obscure destination, did you check to see if there were vans available in Oslo, Lillehammer, or Gothenburg? Did you use a car-rental aggregation site, that checked dozens of companies, or just your home favorite company? You also have to worry about being offered a substitute vehicle at the last minute. Can you drive a standard transmission? What sort of car insurance cost are you planning for?

Our main objectives are to see fjords and experience Norway for 5 days (from Bergen to Flam to Oslo) and also to spend 4 days in and around London before taking a transatlantic cruise back to the US that leaves from Dover.

One way to eperience both Norway and London would be to book multiple destination air. With low cost carriers like Norwegian Air hubbing at Gatwick, you can check out US (don't know your location) to Oslo or Bergen via Gatwick. Do the Norway in a Nutshell train/boat/bus to Bergen or Oslo. Then Bergen or Oslo to Gatwick.

And yes, Norway and the Scandanavan countires can make many Americans feel poor, but you do get what you pay.

Thanks for everyone's input. My research is basically beginning here. Cost is not the main issue (though my funds are not limitless!)- but rather whether driving with all of these considerations in mind (vehicle size, one way fees, ferries, parking, etc.) is worthwhile. I believe I have obtained enough info now to decide it is not desirable. (Yes, I was picturing more of an American road trip not knowing all the ins and outs of traveling in Europe!)
Next, I started looking into travel by rail. That may still be doable. (Although I still picture it a little like an American road trip as that's my frame of reference!) We did take the railway from Paris to Barcelona on our first trip to Europe- so I have a tiny bit of experience to picture. I have not prices out rail travel with the multiple stops and hotels, nor have I priced out simply flying to London.
There are so many options of places to visit and ways to go about it that I am overwhelmed and excited for sure!
We do have the luggage and laundry thing under control!
I think I'll move onto looking at just doing Norway and then flying to London and touring Britain and Ireland, It will depend on the cruise itinerary (if it has stops in Ireland), so I'm going to wait a couple of weeks to come back to this planning. The cruise itinerary should be released soon.
Thanks again!

You are wise to consider the train (or flying) over driving a large van, with all the complications. Especially if you want to visit cities, where traffic and parking can be a nightmare.

We have happily traveled around by train as a family group of nine.

I meant to suggest that you consider making your way from Norway to Copenhagen by ferry. Spend some time there before flying to London.

Norway to Copenhagen by ferry

yes - my recommendation too. Look at https://ferry.dfdsseaways.com/Travel?routeOut=OSCH&locale=GA&SalesOwner=17&depDateOut=2019-09-06&routeHome=ONE_WAY&adl=9&chd=0&inf=0&vty=NO_VEHICLE&pet=0&Product= for more information. The prices for 2019 are not published yet, but you can get an idea by looking at the same period this year.

I will chime in too and suggest narrowing your itinerary to just a few places. The logistics and costs of moving that many people often will be quite intense. Hotels are very strict in Europe about guests per room, and few have rooms that will accommodate more than 3-4 people. You'd be better renting an AirBB that can accommodate everyone and has laundry, then commuting and daytripping. A van then would work well. You wouldn't need to worry about luggage or one-way drop off fees. Taking the ferries in the Baltic is also a great idea; the kids will love it and it solves transportation/accommodations/meals in one package.

And I should clarify that I am suggesting the Oslo to Copenhagen ferry as walk-on passengers, without the van.

We once drove a rental car from Norway to Copenhagen and the drop fee was $500. That was twenty years ago, so I imagine it is higher now.