I beg to differ; I wouldn't use London as a base because the congestion of "commuting" to the outlying towns you want to visit would be a hassle. Also, lodging and food in London tends to be more expensive than in some other towns.
With the geography of the places you list, I'd look for a place to stay in Oxford. Being a university town, I would guess it has good WiFi all over as they are used to having visiting scholars who need to work remotely. Oxford is an easy day trip to both S-u-A and Bath (one day each, obviously -- not both on the same day).
As you are a family, I'm guessing you plan to rent a car, which will give you flexibility and will probably be cheaper than buying individual rail and/or bus tickets for family members.
Stratford-upon-Avon: Be aware if you drive up the M40 to SuA, the exit for SuA directs you to a Park & Ride and shuttle bus into town, so they want tourists to go there. We followed the A46 from there which was a mistake; had to drive through congested traffic for a good half hour to end up at the multi-level parking garage right by the TI where the HoHo bus leaves. If you take Warwick Road (A439) to the left off the A46 it is much less congested and runs more or less directly to aforementioned parking garage.