You’re right–the character lost any English accent by the time this story takes place, so British pronunciation is out. Thus he won’t be saying 'ants or 'ampshire.
The names of English counties do occasionally crop up in speech here and seem to be evenly split between the “shyer” and “sheer” pronunciations. The “sher” pronunciation never occurred to me, but may be a good one to consider also.
If I (a Canadian, as is our character, for all intents and purposes) was faced with reading “Toronto, Ont.” out loud as a part of text, I’d automatically read it as “Toronto, Ontario.” But I don’t know how this would work with English place names. How would this character would read “Swanmoor, Hants,” especially if he had heard his English parents speaking about it throughout his life? I hope you can see where my question is coming from.
I must admit, I’m learning from the responses so far. Thanks, folks, and I’m looking forward to more.