Will they relocate my FTTP box?

I'm about to move into a house where the FTTP Openreach box is installed in a location that's possibly not ideal. Upon moving or starting up a new account with an ISP at this house, is there any chance an engineer could relocate the install? Possibly even drill through a different part of the front wall?

I would expect that you would need to order a ‘shift of ONT’ via your new service provider.

Could you not just run a new Ethernet feed from the current ONT location to where you require the router to be ?

Yep. And the current router might possibly be in a good position, I'm not sure. Need to get a builder in to see which wall is easier to run a cable behind. However if I had to run an Ethernet feed it's wouldn't really be ideal. Would need trunking.