Customs at Gare Du Nord

If going to Luzern, why not fly London to Zurich and rent a car there? It will already come with a Swiss vignette sticker attached, which otherwise you will need to purchase at the Swiss border (40 CHF), not to mention tolls and gas and time. And a lot of fun driving in Paris.

There is no customs between the UK and France because both are in the EU.
There is passport control (immigration), but French immigration takes place at St Pancras before you board the train. This is one of the reasons why you have to get to St Pancras and be through the ticket barriers at least 30 minutes before departure.
When you get to Paris you just walk straight off the train.

I can't say how long it takes to get a hire car, I have never done it there.
But, if you want to go to Luzern the same day. It would be a lot quicker to do it all the way by train. You can get another train (high speed TGV) from Paris to Basel, and a Swiss InterCity from there to Luzern. I have done London to Switzerland by train many times.

For more info on how to do London to Switzerland by train see here: http://www.seat61.com/Switzerland.htm#Zurich

P.S. You don't "take the chunnel". The real name is the Channel Tunnel, and you can't take it anywhere, as it is doesn't move. You are taking a Eurostar train, which goes through the Channel Tunnel for one part of the journey.

Will French immigration be lengthy in London?
We have looked at trains and planes to Switzerland and for four of us it gets expensive. We are also going to Italy so a car just seemed more practical.

The Eurostar requires you be checked in 30 minutes before departure. So one hour is safe. They will look at your passport and stamp it, and a security check, not quite as stringent as airport. That's it.
http://www.seat61.com/London-to-Paris-by-train.htm#Eurostar check-in

If you made up your minds on the car, so be it. But I hope you didn't rely on Eurail pass prices or the RS rail price estimating maps.
Also your price estimates for a car take into account fuel prices ($6-$7/US gal) French tolls, Swiss vignette, Italian tolls, Italian ZTL's, robo speed trap cameras, return your car to France to avoid big international drop fees, need for International Driver's Permit. I think that covers it.

If you buy train tickets early enough it would probably cheaper than car costs especially if you plan to turn the car in in Italy. As others have noted, there is no customs issue at all and no immigration issue in France -- that is all taken care of in London when you go to Paris and in Paris when you go to London.

Passport control takes a lot longer in London than it does in Paris.

The British agents ask a lot of questions, but the Parisian agents just glance at your passport, stamp it and you're out of there fast.

As the others have mentioned, you'll clear French Passport control at St. Pancras in London. Assuming you don't have anything to declare, you won't have to deal with "Customs" at Gare du Nord. You'll also have to go through an airport-style security at St. Pancras before boarding the EuroStar.

As you probably read on the EuroStar website, you MUST check-in at least 30 minutes prior to departure. I'd suggest adding some time to that, and perhaps even make it 60 minutes.

Is there a particular reason you're renting a car for the trip to Lucerne? It will be much faster to use the TGV, which travels at up to 300 km/h. In addition, if you're renting the car in France, you must ensure that you obtain the highway tax vignette for Switzerland (or risk hefty fines). It's also recommended that each driver to have an International Driver's Permit for driving in France.

Passport control takes a lot longer in London than it does in Paris. The British agents ask a lot of questions, but the Parisian agents just glance at your passport, stamp it and you're out of there fast.

It may be true that clearing British immigration takes a lot longer (I've never done it) but it takes place in Paris so that's where it would "take a lot longer," not in London where you go through French immigration. Whichever way you travel, when you get off the train, you just walk through the station as you would any train anywhere.

Not exactly.

Going to Paris, all the formalities are before you go up the travelator and board the train. When you step off the train in Paris you step onto a normal train platform and then as you reach the front of the train you just pop out straight into the middle of Gare du Nord.

Going to London it looks like you have completed all the formalities when you go down the escalator and board the train. But the keen-eyed among you will have noticed that among all the officials you saw there weren't any UK Customs officers (you have to look at the epaulettes). When you get off the train in London you are on a secure platform; you still have quite a walk. You stay in a secure area and make your way through the long passageways and escalators and then eventually through the frosted doors and into the station. While you were in the secure area some of the walls are one-way glass with both Customs and Immigration officials behind checking out all the passengers as they walk by. Most are left alone, but the occasional person is stopped.