I'd like to understand how eBay determines that I cannot bid on/snipe an auction under the "seller has restriction on shipping addresses" system. I travel internationally and have home addresses in both the U.K. and the U.S. If I set my default ship-to address to, say, the U.K, is this what eBay uses to determine if I can bid, and therefore snipe, on a "U.K. Bidder Only" auction even if I happen to be in the U.S. at the time?

Usually it is determined based on your shipping address set as the default one on your eBay account.
If you expect an item to be shipped to a different address, please make sure you set it as the default before snipe kicks in. Your manual bid would not work too unless you change it.