Trains from London

You can use Oyster card or contactless payment for trains from London to Bletchley and Hampton Court. No need for advance purchase.

You can see the price difference in standard tickets for Bath if you go to the National Rail website. Compare the prices for tomorrow with that 2 weeks from now and for your desired date of travel

You can only use Contactless (not Oyster) to Bletchley.

But if you pay by Contactless you can't use a 2 for 1 offer at Bletchley Park, if that was your plan. In that case you would need to buy tickets- virtual or physical.

For Bath- looking tonight for Monday almost every train has Advance (train specific) fares of £40.50 each way (if you booked weeks ahead that would be £30.50 each way). As long as you are willing to be tied to specific trains that sees to be your best option. Whether that would still be true in mid April with the Easter vacation happening round about then is speculative- as more people will be travelling. Oyster is not valid that far west.

If you travel outward after 1030 weekdays or any time at weekends to Bath and buy for all three on the same ticket it is cheapest to buy a Super Off Peak Return, with Groupsave (you don't need a railcard for that)- £151.50 for the three of you.
That can be bought on the day, on line or at the station.
That allows you to catch any train on the way back BUT note the outward time restriction

You can do London Waterloo to Hampton Court using Oyster or Contactless, but note Bletchley (from London Euston) can only be done with Contactless, Oyster is NOT accepted to Bletchley.

Bath is the only one worth booking in Advance if you can get cheaper Advance Single tickets, usually going on sale 10-12 weeks out but less at weekends/Bank Holidays.

Thank you for all the valued input.
We thought we would book Bath, and the return train right away, but gave up. Starting with the train, should we book with National or GWR?... from Tottenham Ct Rd or Paddington? We chose GWR and Paddington, 66 pounds @ return,

We were ready to pay for the tickets when we were instructed to load the GWR app as that's where the tickets would be transferred (or pick them up at the station). Loaded the app, but it didn't work. Apparently you have to be in the UK.

We may wait until we are in London and someone can advise us. If it's a National ticket, perhaps we could pick it up at LHR on arrival.

By 'National', I assume you mean National Rail. They do not operate trains - they are the representative organisation for the rail operators. They run a site that directs you to the relevant operating company for that route. They do not sell tickets or run trains.

GWR is the operator for the route you're looking at. You should be able to book tickets on their Web site and have them emailed to you. The app only works in the UK.

Alternatively, you can book that route with any other train operating company of you are having problems with the GWR site. They all sell each other's tickets.

If you've already bought them, just log into the app when you're in the UK, or print them frlm any ticketing machine at the station.

If you book tickets from Tottenham Court Road (TCR) to Bath you effectively pay £3.50 to go from TCR to Paddington. The fare if you pay by contactless or Oyster as a standalone journey is £2.80 Off Peak, £2.90 Peak.

So I would just book from Paddington. Whether the 60p premium is worth it to you for your convenience is your choice.

And, yes, you can collect any rail ticket at any National Rail Station including the Elizabeth Line/Heathrow Express stations at Heathrow.

Taken to an extreme you could book a Plymouth to Penzance ticket and pick it up at the machine at Inverness at any time between booking and travel.

Regrettably neither GWR nor SWR transfer tickets to email. Only their app (which don’t work where I live), pickup at the station, or mail within Britain. Too bad, that would be a 1522=60 pound saving for us, almost 20%.

If we choose to pick them up at « the station », do you think we could purchase them now, and pick them up as we get on the train to London at Heathrow? Would thy email us a document that we would show at the station?

Is there a preference to purchase from GWR or SWR?

From GWR: « Our website allows customers outside the UK to register with us and buy tickets. We are unable to deliver overseas, however you will be given the option to collect them from a self service ticket machine. If you are travelling from Heathrow Airport to a UK destination you will be able to collect your tickets from Heathrow Terminals 2, 3, 4 and 5.« 
Done.
Thank you all.

Picking up at the station is easy. I have done it many times. Just uses the machines in the middle of the concourse with the code they email you and the credit card that made the purchase. Takes only a minute or 2. The savings is worth it.

It is surprising that GWR don't seem to have an e-mail option. And I suppose it is a big ask to trust that the app will work when in the UK.
SWR (if you buy through their website, not their app) do offer a PDF e-mail facility.
And my own company, Northern certainly e-mail tickets. I kind of assumed every train company did.

On GWR, unless they are way behind the times, although you have to select a station for collection- collection should be at any station. Selecting a pick up station is, well, quaint.

The old fall back answer, as good as any, is to use the LNER app.