As pp noted, you can likely walk up. They do take ressies because I made on my last trip, but when I got there, there was no line.
Reservations are by phone: 407-560-0253, but again as pp noted, calling from the UK would be a toll call.
One tip I can give you though, it about 4 weeks from your departure, go to their website
www.bigrivergrille.com and sign up for "BigRiver Online".
They will send you a "Welcome Gift" by email for a free appetizer. The date of the email offer does expire, so don't sign up too early or the email will expire before you get here.
Enjoy the free app.