Paddington Station detail

Can't find an email address for them, but you might try calling and getting the info straight from the horse's mouth so to speak and perhaps they can send you an email reiterating what they say. Try calling the number on their web site as the call shouldn't be that expensive.

The Heathrow Express uses tracks 6 and 7 In the main station area---street level, not the subway level. Your train to Bath will depart from the same level.

You can check their website to be sure, but you should be able to bring the bike boxes aboard and stow them in the luggage area right by the doors.

From the HE website:

"...Bicycles are carried subject to these conditions of carriage. They are not carried on trains leaving Heathrow between 7.30 and 10.00 or Paddington between 16.30 and 19.00 except on Saturdays and Sundays..."

Why not avoid London entirely and take the bus directly to Bath from Heathrow?

I have done lots of cycling in Britain and trains can be confusing. Check the national rail.co.uk website. It shows that most trains accept cycles - both to Paddington and Bath. By booking in advance, you can get about half price off an anytime advance ticket to Paddington. Prices to Bath are more complicated but you should be able to get a discount with anytime advance.
The trains are run Great West whose policy accepts bike but recommend booking your bike in advance, sort of hard to do when flying in. The national rail site both sells tickets with no commission and links to the individual railway carrier. I suggest you follow the links to GW, then use their email link to ask the question. Armed with a copy of their response, you should get on the train.
I would avoid buses as they normally only take bikes if space is available and at the discretion of the driver.
A good website to get specific info on cycling in Britain is ctc.org.uk ( website for Britain's largest cycle touring organization ).
Enjoy
Al

Rather than using the Heathrow Express to Paddington, you can take the Heathrow Connect to Hayes & Harlington, then train to Reading, then train to Bath Spa. It does show an 8 minutes transfer at Reading from platform 14B to 1. I'll let someone who knows the layout at Reading to comment on that idea.

How big is a bike box? What are the measurements?

Lo asked the right question. A bike in a box is treated as luggage, not as a bike. Bike boxes come in various sizes, depending on how far down you disassemble the bike.

Thanks to all. I think Lola's response that the Express comes in a train (street) level answers that question perfectly.

The bike box is approximately 47"x 31"x 11". It is bulky but no larger than many other over sized pieces of luggage. And it looks like luggage. Sounds like the Express has reasonable luggage space. I assume the train to Bath will also have luggage space but it would be nice to know.

I have heard that the buses often refuse over sized luggage but will explore the Heathrow to Bath bus option.

I plan to take the coach from Heathrow to Bath. This coach link -- http://www.nationalexpress.com/Help-and-advice/coaches/luggage-and-lost-property.aspx -- might help some. I didn't explore much, but it looks like a very generous luggage allowance, and I saw nothing about size, only about weight. The luggage will go in the bottom of the coach. You need to look more deeply than I did and maybe email them.

Check your route at www.nationalrail.uk. The Great Western operates the route to Bath Spa. Click on the little bike icon on the details portion and get a quick explanation. Looks fine.

The maximum size for luggage under the National Rail Conditions of Carriage is 90cm x 70cm x 30cm. These bike boxes are bigger than this, so it will be at the discretion of the operator.