I lived in Jackson years back and worked in the hospitality industry. If you are staying in motels/cabins, be aware that it will be pricey even in June or September and you will need to make reservations well in advance - like 90+ days if possible.
Jackson Hole, Wyoming probably has more direct 1-stop connections than Billings, Montana - plus you would be immediately in paradise without having to risk an iffy shuttle for 13 people. Jacskon has direct flights from Denver, Salt Lake, Chicago, and Dallas - - so you would have plenty of options with a London international connection. You would be at 6200 ft rather than 3200 ft in Billings - a recoup day would be prudent not only for weariness, but for altitude.
Speaking of altitude - you can avoid altitude sickness by taking precautions before - but it's tough to knock out after. Plenty of fluids, limited alcohol, caffeine, nicotine, if O.K. an aspirin a day, don't overdo it at first.
Prevailing winds tend to be southwesterly in the region, less so in June, more so as the summer progresses. They are also impacted/magnified by topography. That said, riding westbound from Cody into Yellowstone can put you in a brutal maw for 50 miles - a good part totally exposed. Again, no guarantees with wind, but the Wapiti Canyon usually has strong westerlies. It is a lovely ride - lovelier when you can raise your head.
Of places to stay three nighst in the park on a bike trip, Canyon is waaay better than Grant Village. Both have a range of facilities, but the hiking is so much better around Canyon - along either rim of the Grand Canyon of the Yellowstone, to backcountry lakes, up the Hayden Valley with all its wildlife. There's a slightly longer, paved back route from Bozeman to Livingston - a moderately longer one thru Clyde Park. Then there is the very quiet East River Road south of Livingston.
Unless your riders are extremely strong, I would caution you about riding the Beartooth Highway from Cooke City to Red Lodge. Especially if you are doing this early or late - the Beartooth Plateau makes Yellowstone look tropical - plus you are exposed in near-tundra conditions for a considerable distance. The climb is brutal and the descent is terrifying - melted rim variety.
The nice thing about starting in Jackson would be that you have an initial 50 miles of relatively flat riding, but eye-popping vistas. A great way to get acclimated.