It sounds from your original post as if you may have 8 full, non-jetlagged days in England. That's certainly better than "8 days" that turn out to be 6 real days. I agree that it would be more relaxing and flexible to spend every night in the same lodgings, and 3 days is very short for all that London has to offer. In addition, one could argue that there's more to do (certainly a great deal more variety) on Days 4 - 5 - 6 in London than on Day 1 at your other locations.
On the other hand, it's kind of miserable to find yourself on a day-trip with the rain pouring down on you (precipitation's hard to predict accurately) and no way to make a quick trip back to your hotel to change clothes/socks/shoes. Another issue with multiple day-trips from London--planned at the last minute due to perfectly reasonable spring-weather concerns--is the cost. If tickets are puchased on the travel day or even the day before, there may be limited opportunity to snag bargain-basement Advance tickets, and the round-trip Anytime fares can be quite high: up to £67.40 for Oxford, £203 for Bath, and £77 Moreton-in-Marsh/Cotswolds. By fiddling with departure times you could probably avoid paying the absolute highest prices, but as you can see, this could be big bucks for your group. You can take a look at fares on the NationalRail website.
Because of the logistical challenges, I don't think an independent day-trip to the Cotswolds is a good idea from London. I know there are bus tours, but there would be a lot of time riding a bus as well as a lot of time loading and unloading if a full-size bus is used. I make no comment on the advisability of visiting the Cotswolds in April because I have never been to the UK outside the period of very late May to mid-September.
If you are absolutely wedded to the idea of seeing Oxford, Bath and the Cotswolds and don't intend to bail if the weather is not optimal, I would spend some nights out in that area rather than making 3 probably-costly round-trips from London. There are one-day van trips to the Cotswolds offered from both Bath (MadMax) and Moreton-in-Marsh (GoCotswolds and Secret Cottage). Moreton-in-Marsh is a short train ride from Oxford; I spent several nights in Oxford and easily took the train to Moreton in time for the GoCotswolds tour.
I don't know whether there's a central base for Bath-Cotswolds-Oxford that would work if you plan to depend on public transportation, and you'd probably need two cars if you chose to drive. Folks on this forum have pointed out that it's difficult to find a vehicle that will contain 7 people and their luggage, and in some countries the driver of a suitable vehicle might need a commercial license rather than a regular driver's license.