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.