There is left luggage in the arrivals hall in each terminal; cost is ~£9.50.

I like to run along the A4 (north of the runway) and watch planes taking off/landing.

If you leave your bag at T5 left luggage, you could use the buses in the LHR free fare zone to get to the LHR Sheraton and head east from there on the A4. At Harlington Corner (Holiday Inn Ariel - round hotel), you could turn right and then make the next left to wrap around the east side of the runway.

Be advised tomorrow's weather does not look good right now.

Edit: looks like it's about 3.9 miles from the Sheraton to Hatton Cross Underground station. I'm pretty certain there's sidewalks the whole way.