Tube
The Piccadilly line connects Heathrow Airport to central London and the rest of the Tube system. It is the cheapest way to central London. There are trains every few minutes from approximately 0500 (0550 Sundays) to 2345 (2250 Sundays). A single ticket into central London (Zone 1) costs £4. Journey time to Piccadilly Circus is about 50 minutes. There are two Underground stations at Heathrow Airport, serving Terminal 1-3 and Terminal 4.

Source - http://www.tfl.gov.uk/gettingaround/2558.aspx

You may have an issue with the 5am tube opening (hope not Sunday) and the 50+ minute ride to get you to LHR on time.

HEX might be faster (tho still a cab from hotel)

Monday to Saturday From Until Minutes past each hour
Departs Paddington 05:10 23:25 10 - 25 - 40 - 55
Sunday From Until Minutes past each hour
Departs Paddington 05:10 23:25 10 - 25 - 40 - 55

A shuttle might work http://www.london-airports-shuttle.c...w_transfer.php

I have used Hotelink before but you need to allow a lot of extra time as they pickup at several hotels. http://www.hotelink.co.uk/ tho they do start at 4am