Best is if you call both Betterment and Vanguard and explain your situation to them. I'm with Schwab, had an international account with them when I was in Japan, they converted it into a "US" one when I moved to the US. The process was reasonably painless, and I didn't have to sell/transfer anything.
My point is, Vanguard or Betterment might be fine with having a UK account transferred to the US and vice versa