Do we need to purchase train tickets ahead of time for Heathrow to Bath?

No, you don't have to buy ahead of time. Trains cannot physically sell out.

Buy your tickets from the machines in the Elizabeth Line/Heathrow Express station at whichever Heathrow terminal you arrive at.

If travelling to Bath in the evening weekday peak those trains are very busy, and you may end up standing.

What you can do is to reserve seats on the train you hope to be travelling on without a ticket purchase by creating a free account with Great Western Railway then following the instructions.
If you land early or late and are on a different train as soon as you have Wi-Fi or data go back in and reserve new seats on the correct train.
No trains run direct from Heathrow to Bath. There are 4 possible routed-
Heathrow to Paddington via the Heathrow Express, train to Bath ;
ditto via the Elizabeth line, train to Bath;
Heathrow to Reading changing at Hayes and Harlington, train Reading to Bath;
Rail Air Bus Heathrow to Reading, train Reading to Bath.

Hey isn31c, thanks for all the info! We will be traveling on a weekday morning so hopefully it won't be too busy going that direction. I will definitely look into reserving seats thanks for that tip. And I now see that it is not a direct trip. Is one of those routes preferred?

Does all of the above also apply to the train from Edinburgh to London? We will be taking that train later in the trip after driving up to Scotland.

For Edinburgh to London the same principles apply, you can buy that day, and can reserve seats on the GWR account if you want to.I
For LNER only you can also reserve on their own website up until 5 minutes before travel.
However you will be able to predict what time you want to travel, so you may as well buy that as an Advance (Train Specific) fare. The sooner you do that, on the LNER website the cheaper it will be.
So do that today if you want, that will include a free seat reservation.
If you wanted to be flexible there are almost always higher priced Advance Train Specific fares until 5 minutes before travel, so it is unlikely you would pay the full flexible fare of £204.80.
On that route for a bit more flexibility there is another fare type called Flex 70. It's slightly more than an Advance Fare, but you can use any train up to 70 minutes before or after your booked train. Today on the next train, at 1630, the Advance is £88 and Flex 70 is £108.
As trains are half hourly that gives you a choice of 5 trains.

As for the best route Heathrow to Bath, I would rather go the wrong way to Paddington where the train starts so there is time to find a seat, stow luggage etc.
Time wise between the 4 routes there is probably 30 minutes travel time difference and around £20 cost difference.
So whatever suits how you are feeling when you arrive, and how tired you are.
Heathrow Express is fastest,.probably easiest and most expensive. But for that small a cost delta, does it really matter?

You can also consider the National Express coach (bus) from Heathrow to Bath. There are some direct non-stop coaches that take just over 2 hours.

https://www.nationalexpress.com/en

For your Edinburgh to London train journey you'd want to book that ahead of time for the best prices.

isn31c, thanks again for all of the great information. The Flex 70 tickets sound like they could work out well.

Pam, I hadn't considered taking a bus I will look into that. Thanks!

The National Express buses are quite fine. That would be my choice if doing your route to Bath. Easier than carting luggage around and changing transit in London. Of course, my preference may not be others’ choice.

we will also be arriving Heathrow 6am on a Sat in August traveling to Bath and trying to decide train vs bus
what experiences has anyone had with customs on a Saturday early in the morning? How long should we expect it to take?

The big issue with the direct bus to Bath is that it only runs every 3 or 4 hours. If you just miss one it is a very long wait, when there is a half hourly train.
Just to avoid walking a few level yards between trains at Paddington if on the Heathrow Express.
Landing at 6am there is a bus at just before 8am which you should make If on time. If you don't the next is at lunchtime.
Also the RS guidebook is out of date as Flixbus now also run the route, but at almost the same time as National Express.
Also National Express sell connections at Bristol, claiming that the second bus is non stop from Bristol to Bath. It isn't- it is a local bus which stops something like 40'times between the two towns.

Hello OP, may i know your experience of registering for GWR account or taking the national express bus or train from Paddington to Bath Spa? May i ask how long did it take you to finish the custom check at Heathrow Airport before you exit from Heathrow airport? Thanks.

Hi J@p28al, I still have two weeks until my trip. I'll try to update this after that

Ok noted. Enjoy your vacation to the fullest. Hope to hear from you next month. Thanks.

Updating this with my experiences for J@p28al And anyone Else.

I did end up creating a GWR account and reserving seats on the 10:30 train from Paddington to Bath.

Our flight got in around 7:30 AM on a Thursday. We walked from the terminal to the passport control area (took us about 20 minutes because we walked all the way instead of taking the shuttle train). Passport control was not very busy when we got there, so I think it only took us 15-20 minutes to get through that. We had some coffee and breakfast in the airport outside of the arrivals area. After this, I purchased train tickets from the kiosk, I believe the tickets I purchased were "Heathrow to Bath Spa by any available route". So we took the Elizabeth line train to Paddington. At Paddington you have to go up and out of the underground Elizabeth line station to the main rail station. For some reason our tickets did not work for either exiting the underground station, or entering the main rail station. However, the staff at the gates let us through. We got there at around 10:27 and saw that the next train to Bath was leaving at 10:30. So we made a dash for platform 3 and made it without a minute to spare. I realized after boarding that this was actually the train that I had booked seats on.

Hi jay,

Thanks so much for your reply.

(took us about 20 minutes because we walked all the way instead of
taking the shuttle train).

Really? The shuttle train? I'm not aware of that. Maybe i do not notice it. I try to be on the lookout for it.

May i ask how long did it take you to arrive from Heathrow airport to Paddington via Elizabeth line?

How much did you pay for the train ticket from paddington to Bath when you bought the tickets from the kiosk machine on that day?

Hey J@p28al,

Really? The shuttle train?

Yeah it's just the little train in the airport that takes you between the terminals. You will see it when you are leaving the terminal. There was a long line for it so we thought we would just walk, but the walk turned out it be a lot longer than we expected.

how long did it take you to arrive from Heathrow airport to Paddington via Elizabeth line?

I believe it was around 15-20 minutes.

How much did you pay for the train ticket from paddington to Bath when you bought the tickets from the kiosk machine on that day?

Looks like I payed $185 (USD) for the two tickets from Heathrow to Bath

I believe the tickets I purchased were "Heathrow to Bath Spa by any available route". So we took the Elizabeth line train to Paddington. At Paddington you have to go up and out of the underground Elizabeth line station to the main rail station. For some reason our tickets did not work for either exiting the underground station, or entering the main rail station. However, the staff at the gates let us through.

Backtracking on train tickets is not normally allowed for a normal routing. You went into London, and Paddington on the very same tracks you came back out on to go to Bath, passing not far from Heathrow. The gates would not allow such a routing. The staff were kind. They won't always be so.

The route it would have expected would have been Elizabeth Line one stop to Hayes & Harlington, change direction and Elizabeth Line to Reading where you pick up the GWR train to Bath Spa.

I am sure you wouldn't have been the first person the gateline staff would have seen on that ticket.

If I am wrong I am sure Stuart will correct me.

Nigel, that is interesting. So if I wanted to go via Paddington (to avoid the extra transfers) I should have just purchased a separate ticket?

I didn't think twice about Jay's comment. As far as I know this is an unusual route where backtracking is allowed- it is if you use Heathrow Express from the airport to Paddington. So I would have thought it was via the Elizabeth Line.
Aside from the fact that it would be cheaper to split the fare- LHR to Paddington then Paddington to Bath- if I had bought an any reasonable route ticket I would have done the same thing as Jay, rightly or wrongly.
So maybe I would have got caught by eagle eyed ticket inspectors.
The ticket gates can just be capricious.

One example I know to be true- if I book from Cumbria to Watford Junction the usual route it offers me (even on a cheap Advance Fare) is Carlisle to Euston, then back out to Watford Junction. It doesn't ask me to change at Crewe and/or somewhere else (Rugby, Milton Keynes or Birmingham) to avoid London.
I don't think it was always thus, but it is now.