Hi Neil,
Ranking for a person's name in the US with a 'name.co.uk' is possible - for example I think if you do a search for 'Jane Copland' in the US you'll see janecopland.co.uk ranking 1st.
Do you know how well or otherwise the site is ranking in the US right now?
Essentially the ability to rank will likely be impacted by how common or otherwise the person's name is and how strong their site is.
In terms of actions from you I wouldn't be inclined to change hosting as that's unlikely to have an impact on it's own - however if as you say you're looking to change the hosting in any case then moving it the US probably wouldn't hurt. I'd also encourage you to build links to this domain to strengthen the site.
I guess I'm saying it's not necessarily impossible, but in an ideal world you'd get some sort of generic domain (.com / .net etc).
I hope this helps,
Hannah