Purchasing a Eurail Pass and then getting reservations

If these are the only trips you'd like to take and you have a totally fixed itinerary that long ago, you may check normal ticket prices, too. 

If you do buy a rail pass, you would just need two reservations (London to Paris and Paris to Luxemburg) so it's maybe easier to do for you. Benefit: You are much more flexible regarding departure times (recommended).

You won't need a travel agent for this. It's easy to book. I'll give you both options (normal tickets and rail pass reservations):

Southampton to London:

Normal ticket: Buy Southampton (Central) to London (Waterloo) here: https://www.southwesternrailway.com/buy-train-tickets/ticket-search-results?openQTT=1 (it's already possible to buy but it will get much cheaper if you do wait for about three more months and book then - a good price is about 10 £, don't buy it if it's still 52 £). 

Rail pass reservation: It's not possible to reserve a seat for this. If you buy a rail pass, you may just hop on the next available train (every 30 minutes). I did arrive by Queen Mary 2 a few months ago and made it for the 10am train without using a taxi.

London to Paris:

Normal ticket: Buy London (St. Pancras) to Paris (Gare du Nord) here: https://www.eurostar.com/us-en/train/france/london-to-paris (already possible and should be done now to get a cheap fare).

Rail pass reservation: If you decide to buy a rail pass, you may reserve a seat (expensive, it will cost 30 € 2nd class or 38 € 1st class) here: https://www.b-europe.com/EN/Booking/Pass#TravelWish (should also be done early).

Paris to Luxemburg:

Normal ticket: Buy Paris (Gare de l'Est) to Luxemburg here: https://www.sncf-connect.com/en-en/train-ticket (not yet possible but you may provide your Email when searching for your connection and you'll get a notification when it's possible to book).

Rail pass reservation: With a rail pass, you should call SNCF when the notification comes and book by phone (https://www.sncf.com/en/customer-service/contact-us/telephone) or book a train to Metz here: https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish and change for a reservation-free regional train to Luxemburg).

Basel to Luzern / Luzern to Zürich:

Normal ticket: Buy Basel (SBB) to Luzern and Luzern to Zürich (HB) here: https://www.sbb.ch/en/buying/pages/fahrplan/fahrplan.xhtml (possible two months in advance).

Rail pass reservation: If you decide to buy a rail pass, there's no need to reserve a seat for these connections - just go to the station and board the next available train (they do run very often so it's even more comfortable to take just the next available train wihhout need to wait for a reserved one).

If we decide to use a rail pass do we use Eurostar or Eurorail?

Thanks

Eurostar is the train (company) connecting London to Paris, Lille, Brussels, Rotterdam and Amsterdam. Eurostar has nothing to do with rail passes.

Eurorail is unknown to me. Do you mean Eurail? It’s the only organisation that provides all encompassing train passes for most of Europe.

If you decide to buy seperate tickets, you can buy tickets on all websites mentioned above by @Hektor or use a website like https://www.thetrainline.com/ where you can buy all tickets of your planned itinerary.

Although I prefer buying with the companies themselves.

Unless you value the flexibility, I don’t think a pass offers great value in your case.

If you’re happy to commit to your trains in advance (a few months for London-Paris and Paris-Luxembourg, a few weeks for the other journeys) you can do this for about half the price of a Railpass + reservation costs. And it’s just as easy.

Even if you have to buy a walk-up fare for some journeys, it would be hard to hit the $275 it’s going to cost with a pass.

For example Lucerne-Zurich is only 25 CHF on the day, or maybe half that if you book early.

If it was me, I’d buy an afternoon Eurostar from London ASAP for £74. You can change time and date (but not change route or get a refund) later for free. Just pay any difference in fare - up to 14 days before departure.

Then I’d buy the TGV from Paris to Luxembourg as soon as reservations open. Your date isn’t available yet, but there are some the week before for €50 (and just €5 extra for first class).

Then I’d wait until closer to the time, to buy my ticket from Southampton and the Swiss tickets.

As above, Trainline is really good for quickly finding prices, but it’s not always the best place to buy. Usually better to buy direct.