I have an early morning (6AM) flight to Cairo out of Heathrow.
I looked up the Tube (first departure is at 5:30AM) and Airport
Express (first departure is at 5:10AM). Taking either of these
will jeopardise my departure to Cairo.

I am on a budget and would like to avoid taking a taxi or staying
in one of those expensive hotels near the aiport.

How can I get to the airport in time?

1. Are there any cheaper hotels near Heathrow (around US$50)?
2. Is there any other mode of transport that starts around 4AM?

My only other option is - get to the airport by mid-night and wait
there until the morning.

Due to the recent developments, is it allowed to stay at Heathrow
overnight?

Thanks