The walk from T4 to the tube station is down either an escalator or lift then along a flat corridor to the ticket office and platform.
Victoria will be easier to reach but there are stairs at the Victoria end. Piccadilly line to Hammersmith then change across the platform and take any District line service to your final destination. Allow about 50 minutes to an hour.
To get to Waterloo you can stay on that District line train as far as Westminster where you would need to change to the Jubilee line. This is done by escalator although there is also a small lift. An escalator will then take you to street level at Waterloo.
You can do it just as quick and avoiding central London
Take the X26 bus from the Central Bus Station (you'll need to take the free shuttle train from T4 first) to Worcester Park Station, then a train to Epsom
Another alternative if you don't like buses - tube to Earls Court (some steps can be avoided by changing at Hammersmith), tube to Wimbledon (step free station) and train to Epsom
I think Alan's suggestion is actually more effort and might take nearly as long bearing in mind that trains only run every 15 minutes from T4 to the main station and then there is a fiddly walk to the Central bus station. The X26 only runs twice an hour and takes almost an hour to reach Worcester Park.
If the timings were right and you did want to take that bus route then I would actually take a free 482, 490 or 555 bus from right outside the arrivals hall at T4 to Hatton Cross to change to the X26. Everything is on the flat and there is a small newsagent selling drinks and snacks should you have to wait.
Another 'simple' alternative avoiding the crowded Underground and Victoria Station etc is to take the 490 bus from T4 to Feltham Station, journey time about 15 minutes
From Feltham Station you can get a single ticket to Epsom changing trains at Clapham Junction which should take no longer than an hour.
All the frequent trains from Feltham go to Waterloo but by getting off at Clapham Junction you can pickup the trains coming out of Victoria and avoid Central London all together.
Having said that at Clapham Junction you wll need to be prepared for dragging your luggage up/down stairs when you switch platforms.
But it's much more interesting to take a tour of south London than be stuck in a hole in the ground
At what time are you arriving at Heathrow, and where in Epsom (hotel, address or street name) are you heading to?
There are 4 trains per hour from Waterloo with a journey time to Epsom of 33-37 minutes, and 2 tph from Victoria with similar journey times. Other departures from from Victoria take an hour.
To get from T4 to Waterloo, either (a) take the Piccadilly line tube to Green Park - sit in the rear of the train - then switch to the Jubilee Line by taking the escalator up to the ticket hall, then two shorter escalators down to the Jubilee line, for a southbound train to Waterloo, or (b) Piccadilly line to Hammersmith, for a very easy cross-platform change to the District line to Westminster, for lifts down to the Jubilee line to Waterloo. Both journeys have no stairs.
The bus to "Feltham, train to Clapham Junction, then change" route is one I'd like to try, but I suspect it will take longer than the tube then train. As of last month at Clapham Junction, the Epsom platform (11) is one of the remaining platforms not to have lift access from the footbridge (though it's not far off being ready), so you'd have 27 (temporary) steps down with your bags. This may have changed since then, or by the time you make your trip, but as of 4 weeks ago that's not an easy change.
Epsom station is one stop outside the London travel zones. Depending on your exact destination (it may be easier to go to another station inside zone 6) and time of departure from Heathrow will determine the best ticket.Edited: 25 April 2011, 09:49
Thank you all for your suggestions. I am arriving on a Wed at 1.45pm.
It appears that the best option (having to lug a luggage) is to use the tube and the train stopping at those stations that have lifts. Thus I will take the tube from Heathrow to Earls' Court, change to District Line to Wimbledon, then take the train to Epsom.
>Thus I will take the tube from Heathrow to Earls' Court ...<
The lifts from the Piccadilly line at Earls Court are closed for maintenance. You would have an escalator ride up to an intermediate level and then 25-30 steps up to the District line platform. You can do this step free by making an additional change at Hammersmith from the Piccadilly line to a London bound District line train (that change is very easy on the same island platform). Then at Earls Court there are working lifts to change to the other (out of London) island platform.
>Lifts closed at Piccadily/Earls Court....<
is there a way to check if the lift at Earl's Court is working again ? I arrive on May 18