taking the tube into London from Terminal 5 Heathrow

The Transport for London site says that there is an Underground station at Terminal 5. Take the Piccadilly Line to Hammersmith, walk a few feet to the other side of the platform, and take the District Line to Victoria.

If you needed to travel between terminals to catch a connecting flight, then you would take a free ride on the Heathrow Express.

The advantages of the tube from Heathrow are cheapness (not only are the standard fares lower, but you can use a six-zone London Travelcard on it) and the fact that you may not need to change trains as much for some destinations elsewhere in London.

However, possible problems:

The Piccadilly Line is deep level with long escalators to the surface at many stations which may be difficult with heavy luggage. Relatively few London tube stations have elevators.

The journey can be quite long because the trains stop at many local stations between Heathrow and central London.

Depending on what time of day it is the trains may get very busy with not just airport travellers but commuters from the suburbs. Conditions are likely to get particularly nasty if you are leaving Heathrow between about 7 and 9:30 am.

Follow Tim's advice. Jump on the Piccadilly line at Terminal 5, change at Hammersmith, and hop on the District line to Victoria--literally across the platform.

By changing at Hammersmith, you avoid the stations with escalators as Philip pointed out. And I can attest. Some of those elevators seem to take forever.

Total time about an hour.

Taking the Heathrow Express or Heathrow Connect truly is out of the way and I doubt you'll save that much time since the tube runs more often than either of the two aforementioned trains. And by the time you walk from the train platform at Paddington to the tube station below, all that extra time is lost.

Save the money for a welcoming pint.

If we take the Piccadilly line from Heathrow to Kings Cross, will we need to use a lot of escalators to get to the platform for the train to York? Or is there a lift available? Thank you.

Kings Cross doesn't have an elevator at the moment. They're putting one in right now, but it'll be a long time until it opens.

Even easier, because there is no crowd, is to get off the Piccadilly line a little earlier at Barons Court. We always used this transfer spot instead. No crowds, as I said, and only about 10' across the platform to pick up the line to Victoria.

Piccadilly line to Earls court station, then take the District line to Victoria station. Earls Ct station is one of the few with elevators/lifts.

Thanks for the awesome advice, the tube it is!