If its an invoice submitted but unpaid then whether the invoice is not due, due or overdue is immaterial at this point.
You would record it in accounts receivable USD as its not been paid.
The currency is USD until the time its converted, that is when you will know the Value of GBP you have. At that point either its converted and you're paid or converted and still not paid. That is when you convert it in your accounts and transfer it to the appropriate account.
But you will only do it at the time you know its been converted, not before, if they do it, great, if not you need another account for currency conversion. Im, not sure how you would work that one out or how it works in your software, but hopefully they will pay you in GBP and eliminate the headache.
In short, Follow the money, its not moved so neither do your accounts.