Are you thing a guided tour or self-guided (lodging is booked for you and you are provided maps and route descriptions to ride on your own)? Mac’s Adventures would be a good choice for this. Here is a sample of their standard Cotswolds tour itinerary, from a base in Moreton-on Marsh:
https://www.macsadventure.com/us/tour-1302/cycling-in-the-heart-of-the-cotswolds/
This has 5 days of cycling, Ut if on any given day it is pouring rain you could choose to hike that day instead. And of course they could customize the itinerary to suit your group. Meals are on your own.
The one drawback I can see with this is that the lodging they book tends to be small B and B’s which might not have room for your whole group, so you would be split up. But they may have places large enough for all of you as well.
For a guided tour, here is a good option with Saddle Skedaddle, a UK cycling tour company:
https://www.skedaddle.com/uk/cycling/holiday/Leisure_Cycling/UK_-_Cotswolds_-_Bourton-on-the-Water_-_Guided_Leisure_Cycling_Holiday/826/view.rails?_gl=1*2vb87*_up*MQ..*_ga*MTc3MjgzNzAwNC4xNjk4ODYxNDc4*_ga_MK0WLVZNPL*MTY5ODg2MTQ3OC4xLjAuMTY5ODg2MTQ3OC4wLjAuMA..
They offer 3 days of guided cycling in their standard itinerary, but for your group I am sure they would be happy to customize. The lodging is a large manor house in Bourton-on-the-Water managed as a hotel by HF Holidays. (This is a member-owned cooperative that owns or leases such houses all across the UK, focused on walking holidays (“hiking” to North Americans)). Full board is included, with breakfast and dinner at the hotel, and a packed lunch provided to order.
We have stayed with HF Holidays on walking holidays in two of their country houses, in the Lake District and the Jurassic Coast. The quality of the accommodations and food was excellent in both cases.
For your hiking, you could either move to a different village or lodging and plan the hikes on your own. Or you could time your cycling to connect with one of the HF Holidays walking programs based at the same location:
https://www.hfholidays.co.uk/holidays-and-tours/3-night-cotswolds-guided-walking-holiday?format=pdf&vid=3177
Here is more information on Harrington House and the surrounding area, together with a link to holiday program options at this house
https://www.hfholidays.co.uk/country-houses/locations/bourton-on-the-water-cotswolds
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