Dont know if this link will work - it may expire
http://www.alitalia.com/GB_EN/BOOKIN...etDetails.aspx
Alitalia have a flight out of Heathrow 9th June 06.50am arriving Rome 10.20am, depart Rome 11.35am arriving Cairo 15.55pm.
Leave Cairo 13th June 4.25am arrive Rome 07.05 but not leaving Rome for Heathrow until 17.15 arrive Heathrow 19.05. You could see if their is another return with another airline with better times. This return is 293.30
There are always flights from Cairo to Hurghada.
Egyptair have direct flights from Heathrow 9th leaving Heathrow 09.15 and arrive Cairo 16.10. Depart Cairo 13th at 17.00pm and arrive Hrow 20.25 for 398.40.
Their website says they have a special offer for Hrow to Cairo return flights for £273 so you could give them a call about these - see their contact page for tel numbers in Heathrow.
There is a flight from Cairo to Hurgh with Egyptair leave Cairo 20.30 and arrive Hurgh 21.35. Then leave Hurgh 13.35 arrive Cairo 14.40 giving you plenty of time for the 17.00 back to Heathrow - cost 72.50.
Alot of mucking about tho. I will keep searching for direct flights.
Hope this helps.