Where to research and buy train tickets that are multi-country?

A lot of question but will try for simple answers. The German train site is a good site for tracking all trains in Europe --dbahn.com. You can only buy tickets on the site for trains starting or ending in Germany. The best site for purchase is general the railroad site for the countryyou are starting from. If going Italy to France, buy the ticket on the Italian site where you are beginning.

You can buy refund tickets. However, the refund rules will vary from country to country or railroad to railroad. Seat61.com can provide good generic info for train travel in Europe. Seat reservations can be a bit tricky. Seat reservation are often an extra fee but in Italy all tickets (except Regionale trains) come with a automatic seat reservation.

And sometimes a refundable ticket is only credit for the next trip. I would not buy many tickets with the idea of frequently changing the ticket. Trains rarely sell out so very early purchases are not necessary. The only reason to purchase tickets early is to take advanatage of discount that could be available.

as far as to and from London, there is only one train company at the moment but several others have made noises that so far have amounted to nothing. The one company is Eurostar. They go from London to Paris, London to Lille and Brussels and onwards to Amsterdam. They have recently merged with Thalys and given their name for various high speed trains in BeNeLux and France and into Germany.

You need to look at each country independently and understand who runs the trains there, and then look for international trains.

When is the trip?

Is the journey from London to Luzern in a straight line? Will you be stopping on the way? Doing anything in the vicinity?

How are you getting from Luzern to Colmar? Same question about on the way...

Colmar to London is straightforward by train. Local French (SNCF - TER) train to Strasbourg and change onto the high speed TGV Est nonstop or nearly so into Paris Gare de l'Est and a 15 minute or less walk next door to Gare du Nord for the Eurostar to London. Unless you intend to stop off on the way?

Fares on Eurostar only go one way - up - and they are high enough for nonrefundable. Refundable will eat your lunch.

How many is "we"? Mobility or age issues?

We are going straight to Lucern from London, no stops. And straight from Colmar back to London. We don’t have enough info yet, so we are not totally sure.
There will be two or three of us. No mobility issues.

I will reiterate Frank's suggestion to look at Seat61.com to learn about various train routes, how and specifically where to buy tickets, what stations and trains look like, etc.

I would second Frank's rec to use Deutsche Bahn, the German railroad company. The English site is here: https://int.bahn.de/en and you can search for tickets and prices in all countries.

I also agree about the Man in Seat 61, which is an excellent resource for train travel. I still enjoy going there just to look around and see what's there. :-) For example, I recently found out that you can find where all the luggage storage places are at each train station. https://www.seat61.com/luggage-lockers-at-stations.htm

https://www.thetrainline.com is an easy site to use for European train travel.

Also consider flying from Britain to Basel/Mulhouse Airport & taking a local train to Basel from where direct trains go to Lucerne/Luzern. You can then go back by train to Basel and another train to Colmar before returning by local train to the Airport.

https://www.skyscanner.net will show you who flies from London (any airport) to Basel/Mulhouse. Note that low cost airlines charge extra for baggage in hold. Don't click through from skyscanner to book but go direct to the airlines' site.

My go to for quick answers on routing etc on any rail journey in Europe would always be De Bahn (German Railways)- https://www.bahn.de/en

That tells me that London to Lucerne is a 3 seat ride (which I already knew)+ the RER across Paris from Paris Nord to Paris Gare de Lyon, and takes just over 9 hours in total, final change of train at Basel SBB.

Departures at 8am, 9.30 am, and 11. 30am (the last one to arrive in Lucern at a decent time of night). My only thought is that typically they allow one hour to cross Paris- sufficient but if the Eurostar is late not over generous.
I would want to allow two hours- just in case.
The 9.30am and 11. 30am actually have better connections in my opinion by taking a TGV from Paris Est (walkable in 10 minutes or so from Nord) with changes in Strasbourg and Basel. Yes an extra connection, but a connection where I can walk between Paris terminal stations.

Colmar to London as said is generally a 3 seat side (about a 7 hour journey)- Colmar to Strasbourg or Mulhouse, Strasbourg to Paris Est then Eurostar, BUT there is a 1000 TGV direct from Colmar to Paris Est- with a two hour connection in Paris.
HOWEVER this morning that TGV is cancelled, as is the earlier 0823 from Colmar and the 0806 TER train from Colmar is 40 minutes late (yes, DB tells me that). Not a good morning on that route for whatever reason.

I would book both journeys through SNCF, although Swiss Railways is another option from Paris to Lucerne.

Note that the 10 am has an alternative connection to London arriving 3 minutes earlier using Eurostar Red (Thalys) to Lille, then a Brussels to London Eurostar. That has its pluses and minuses.

London - Zurich by air starts at less than 2 hours and less than $35. Then another 45 minutes on a train to Lucerne.

Thanks everyone so much for all this advise. It is so helpful.
I am wondering, Mr E…what airline is the $35. flight on. I was looking just now and it seems like they are more like $300 or more per person so just wondering where else to look.

Regarding the flights from LON to Zurich the answer is Easyjet from London Luton or Gatwick. That fare is before baggage fees, and will be higher on some dates, being demand driven.
Add time to the airport and check in, and door to door time is more like 6 hours.

Yup, that's the one. I just picked a random day in late May using Google Flights.

Not more than 6 hours with the flight vs not less than 10 hours using a train alone? I presume most dates you can check a bag and still fly for $100.