I was looking for similar solutions a while back and it turns out it's just not entirely straightforward to do this. You need an external resolver to get your WG tunnel initiated, but then you could switch to your private (i.e. router) as the DNS server. You could script it, if you wanted, but it's not as easy as just setting a dns entry.
I got some good suggestions when I asked, but I never did explore further because of the fact that it was more effort than I wanted to spend at the time.
Worth remembering -- if you use a DNS server that is public (such as cloudflare or google), the traffic containing those dns requests will actually flow through your home router (which is the other WG endpoint), so the local-to-you (travel router side) network provider would not be able to see your dns requests (but your home ISP could). Further, if you wanted to, it would actually be possible to do a dns hijack/local redirect (on the home router side).