Well not much help so far I will try and help if I can,
Stirling is too far from London you will lose 2 days just travelling its a shame because it is a lovely area,
Bath is beautiful it is a lovely place to visit, love;y history and architecture
Unfortunately there is no where cheap in London expect to pay about $200 per night for an OK place, most of the hotels are pretty standard so its really a question of where you want to go the next day, public transport in London is very good big choice or methods, destinations and very frequent. There is a really good bike scheme in London so you can pick it up / drop off whenever you want (just be careful with traffic)
Not sure about Hadrians Wall do you plan to hire a car to get around?
Cross your fingers for the weather, last year that week was 30'C but the previous year it was 5'C but London should be about 15-18'C normally