Heathrow how much time- Terminal 3 to immigration to rail

but he anytime fares are too expensive.

You do realize that if your plane is delayed and you miss your train you will be buying an "Anytime" fare.

No one can predict how long it will take since it depends on how many other flights are arriving. It also depends on what passport you have and whether or not you can use the electronic gates (and hoping your passport works in the electronic gates).

advance rail tickets to Bath...... the anytime fare are too expensive.

I did not look at rail ticket cost, but have you checked out bus fares? Looks like the walk up fare is ~$35-40.

https://www.nationalexpress.com/en

Or book a coach ticket in advance, and add on the £5 option for flexibility, then if you miss your coach you just travel on the next one. You should get that for around £10 + the £5 flexibility add on.
An advance train fare will be around £30, a day of travel fare just under £70.

As you are traveling at the middle of the day you wouldn't need to buy an Anytime fare but a Super Off Peak one of £45.10 (without considering the shenanigans of split tickets etc). This is valid via Hayes & Harlington not London which would be more expensive for little benefit in journey time.

Direct coach is best through if the schedule fits.

I just went the opposite direction, Bath to Heathrow, on National express (coach)and it could not have been easier.

I do realize none of us have answered your original question. With Heathrow, time is variable. How far back on the plane are you? What gate did you use? How many other flights arrive at the same time? How long a wait for luggage? Did my plane land on time or did we circle England in the infamous Heathrow landing pattern? In general, it takes me an average of 45 minutes from landing to leaving the airport. It could be shorter, it could be longer.

One other advantage of taking the National Express bus on arrival is that it is very straightforward. Follow the signs to the coach terminal between terminals 2 and 3. Deal with your luggage by stowing it in the hold and don’t deal with it again until Bath. No transfers to other services to keep track of. Finally, have complete timing flexibility for just 5 pounds more and eliminate this whole timing stress.

No one has explained that you can't catch a direct train from Heathrow to Bath. There are three ways to get there by train: -
1. Elizabeth line or Heathrow Express into London Paddington and then train direct to Bath. The Heathrow Express is quickest just but the extra cost is not worth it in my view.
2. Elizabeth line one stop to Hayes & Harlington, change for a train to Reading and then one to Bath.
3. Coach to Reading for a direct train to Bath.
Choice 2 is relatively new and is probably the best combination of price and speed, but does require two, albeit very easy changes
This is why many people suggest the coach which is direct from Heathrow. The downside of the coach is that they are not as frequent as trains, so it does depend a bit on how the timetable fits with yours.

Generally allow an hour from flight to transport - can be longer, but generally isn't

EL to Hayes/Harlington then more changes will only save 10 minutes so it is a lot easier to EL to Paddington then train from there, one change

Thank you all for the information and ideas.

since the train strikes have started again be sure which day you arrive doesn't match a strike day or a shoulder day. There are postings on this Forum with warnings of strikes when they are announces and with updates.

If intending to travel by coach, the earlier you book the better with forthcoming rail strikes.
The moment a strike is announced NatEx double fares on most routes, I have just been looking at one route where fares have been quadrupled for a strike day.
In fact Bath is one of those routes with fares almost quadrupled on the next rail strike day- so they are now almost the same as the walk up train fare.
Megabus and Flixbus don't do the same thing, but they don't run on the Bath route.