Probability of making train after landing at Heathrow

You haven't really provided enough info here to answer. I'm assuming you're taking the Elizabeth Line train to Paddington and then train to Bath, so the key is not when you take your Elizabeth Line train, but whether you make your connection at Paddington.

Elizabeth Line trains runs regularly, so if you miss one you just take the next one. For Paddington to Bath you could buy an Anytime ticket which allows you to take any train, although they're more expensive than a ticket for a specific service. And peak is 6.30-9.30 for Elizabeth Line and 6.00-10.00 for GWR (the operator of the train to Bath), do not sure what you're referring to in terms of times.

So if you could explain what your route is, that will help.

As a general answer though, you should have time. Customs won't delay you at all - you just walk through it unless you have something to declare. Passport control is egates, and that's normally quick - sometimes a bit of a queue bur not normally a.big delay.

The train station is a fair walk away though, but I think an hour from landing should be plenty of time to make the train, assuming no major meltdowns.

a few things...

If you buy the "cheapest" ticket and your plane is delayed, or you're delayed going through immigration or customs, and you miss your train, you will have to buy a new ticket.

There are other routes besides backtracking to Paddington.

You could take the Railair Coach to Reading and then get the train to Bath. (It's the same train you would have caught at Paddington.)

The Railair coach operates every 30 minutes and it takes just over an hour to get to the Reading Train Station.

https://www.nationalexpress.com/en/destinations/bath/heathrow-to-bath

Take the bus?

I suppose it’s possible to make it to that train but, personally, I wouldn’t risk it. The egates are usually pretty quick but arriving early at Heathrow can sometimes be a longer line to get through. De-planing and the walk to passport control, then the long walk out through a maze of people to the trains/tube after passport control takes some time. And that is assuming your plane arrives on time. I would pay more for a later train or take the bus.

As regards National Express if you are through in time there is an 0750 direct bus, just buy a ticket at the machines if you get to the bus station on time. The next bus is hours later.
If this is after 15 May Flixbus also have a bus at 0740. Buy on line using Heathrow free wifi or your own data.
Taking the rail air coach to Reading won't solve the peak time issue, nor get you there any quicker or cheaper than via Paddington.
The Elizabeth line has no peak time, it is always the same fare.

So if you miss Flixbus and National Express just take the Elizabeth Line to Paddington, get some breakfast then a separate Off Peak (flexible) ticket to Bath on the first off peak train to Bath at 0930 for £63.60.

Something to factor in is the location of your seats on the flight arriving to Heathrow. If you are in business class or even premium economy you should get off the plane fairly quickly once the door is open. But, if you are in economy it can take much longer.

The no checked luggage really reduces the time it takes to clear immigration. The egates make the actual passport control area very efficient. I don't think I have ever spent more than 10 minutes there and usually far less. These are the variables that will impact your timing. Plane on time, Plane allowed to land on time and not placed in the dreaded figure 8 holding pattern, plane allowed to park at the gate on time, deboarding procedures, distance from gate to immigration (almost always a good hike), number of planes processing at egates, time to Elizabeth line.
My journey was in the reverse direction of yours. I went from Bath to Heathrow. I used National Express and was pleased with my journey. Some advantages here. Low cost, can get a ticket that allows for time changes if needed for just a few pounds extra, just walk to the bus station. The driver loads your bags under the bus and you relax on the bus until you reach the destination. No worries about connections while possibly jet-lagged.

I didn't mention the buses because their times were not great.

I arrived at T3 a few days ago. Depending on where your plane parks, it can be a 10 minute walk to immigration. At 6:55 in the morning, it will be crowded.

More time consumers:

Will a gate be available for your flight? Sometimes there is a delay?

Will weather affect your arrival? When the weather is bad, ATC puts more space between arrivals for safety and that eats up time. Or there could just be a lot of arrivals and you can be delayed. (This happened to me on another flight last week where we were kept on the ground at the departure airport for 45 minutes to make it easier for the fight to ease into the arrivals pattern.

Do you have checked luggage?

Will your passport work in the egates? If you have an older passport, it may not work in the egates and you'll have to see a Border Force agent. (Different line than the regular one so it does move quicker. But still takes time.

The Elizabeth Line is not in Terminal 3. It's located via underground passageway between Terminals 3 and 2. Not a long walk but around 5-7 minutes depending on how fast you walk. There are some moving walkways.

Thinking you can make a coach 45 minutes after landing is really optimistic.

This is one of the reasons why trying to buy the cheapest tickets for specific trains or coaches is difficult on the day or arrival. Too many variables.

Thanks everyone, we had considered all of these variables as well and were looking for real world advice and you all have provided that. Seems the answer is that it is too risky so we will wait and take an off peak train a little later in the morning after we grab some breakfast.

There seem to be Advance fares available on trains from 9.30 from Paddington on May Monday mornings. I think you should be OK to book those tickets to save some money.