Obviously exposure to a religion is a pre-condition of converting to it, but I don't think there's a one to one correlation at work here.

For instance, there are a handful of notable British converts to Islam in the age of empire: Marmeduke Pickthall and St. John Philby spring to mind as particularly notable examples. Pickthall in particular was important in supporting Islamic life in the British Isles, including the establishment of one of the oldest mission societies in the UK.

But these men were very much the eccentric exception to a very Christian rule.

It's likewise worth noting that although recent figures are outside the scope of this sub that, simply by way of comparison, conversion in the UK to Islam seems to have most benefited from the presence of immigrant Muslim communities than from occupied imperial possessions.

When I say the correlation isn't one to one though, obviously there has been a pattern of settlement for residents of former imperial colonies to immigrate to that home territory, i.e. Algerians to France, Pakistanis to England, etc.