Hello Bookstack,
I travel via National Coach from Fareham to Heathrow. Drops me off at either T5 or the Bus/ coach station. Done this a few times usually early in the morning, earliest is about 4 am, if I do not need an early flight. I usually book online or there is an office at the bus station in Fareham. In October I will be booking Premier Inn at Terminal 4, a new hotel. It has links with the other terminals. There are a load of hotels on Bath Road with Hopper buses- not easy when you have large suitcases. Have a look online with Premier Inn and National Coaches.
Jan