We're going to land at Heathrow Terminal 1 and need to travel to Bath.
We should get through customs around 4:30 - 5 pm on a Saturday. Should
we get non-refundable inexpensive train tickets? If our flight is
late and we miss our train, how do we deal with it?

For the cost of two full price tickets, we could take a taxi instead.
Will traffic be a problem at that time?

Coming back, we're flying out of Terminal 3 and want to spend the
night near Heathrow. My travel agent suggests the Hilton adjacent to
Terminal 4, but I see that the tube station there is closed for
construction into September and wonder if getting to Terminal 3 might
be just as easy, if not easier, from one of the hotels outside the
airport.

If we spend a day driving and touring from Bath to Heathrow, what
sights or towns should we see?

Any advice would be welcomed. Thanks.