I don't know whether this would actually work out within the allotted day, but here it is (with Ken O done within the window):
Heathrow Terminal 4

Heathrow Terminals 123

Heathrow Terminal 5

Barons Court

West Kensington

Kensington Olympia (28 bus)

Earl's Court

Cockfosters

High Barnet (384 bus)

Finchley Central

Mill Hill East

Morden (via Bank)

Wimbledon (93 bus)

Upminster

West Ham (c2c)

Stanmore

Edgware (142 bus)

Tottenham Court Road

Woodford (via Hainault)

Epping

Leytonstone

Walthamstow Central (257 or W15 bus)

Brixton

Stockwell

Elephant and Castle

Baker Street

Bond Street

Ealing Broadway

North Acton

West Ruislip

Uxbridge (U1 bus)

Ealing Common

Turnham Green

Richmond

Hammersmith (D&P)

Hammersmith (H&C) (foot)

Paddington (Bak/Cir/Dis)

Marylebone

Harrow and Wealdstone

Harrow-on-the-Hill (140, 186, 258, 340 bus)

Preston Road

Rayners Lane

North Harrow (H10 bus or foot)

Watford

Moor Park

One of Chesham or Amersham

Chalfont and Latimer

The other of Chesham or Amersham.
Can anybody think of a lazier one?
EDIT: The inspiration for this route came from Sam Gunner, who during planning posed the question "When leaving Heathrow, is it better to change to the District at Acton Town or get as many stations done as possible by going all the way to Cockfosters?" This intrigued me as to whether a route that avoided all those pesky Zone 1 changes could actually work, whether a route where each line was ticked off one at a time could be the solution. A route
minimising the devastating effects of tube delays and line closures.
Of course when routing it through Journey Planner, or to be more accurate, Google Maps, I discovered what I should have known all along, that it couldn't possibly compete with any sensible route - but it could be the answer for people with no clue as to how to begin.