Coming from the UK, then you definitely need to visit Glacier Park. One of the gems in our National Park system.
A few notes:
- No bikes allowed on the singletrack/hiking trails in our National Parks.
There is a bike path from West Glacier into Apgar
There are restrictions on riding times up the Going to the Sun Highway, do your research on that
But, you can take your bike on the shuttle bus and do the downhill, which is a fantastic way to see the park
So a good trip would be from Banff on the GDMBR to Whitefish and detour into Glacier, seeing the sights and riding out the east side of the park and working your way to Waterton Park (Canadian side). I'd bet you could get easy transfer to Calgary or Banff from Waterton. If you need more days, go from Whitefish on the GDMBR to Seeley Lake and just backtrack to hit Glacier. That part of the trail is so nice, you won't mind riding it in both directions (that is what we did).