neither Arran nor Bodmin (both suggested above) is on any conceivable route you'd take from Glasgow to London. This country takes longer to get around than you might imagine.

I suggested Arran because one of the people travelling had lived in the next island up in the Firth of Clyde, and must know the place (it's somewhere you'll probably want to revisit if you've been there once). There isn't anything in Bute at that time.

Arran is close enough to Glasgow and well enough served by trains and ferries that I've climbed Goat Fell from Glasgow as a day trip two or three times. Hundreds of people do it every day during the summer.