In BrE, you will certainly hear "I've booked my holiday". In the case of a package holiday, you can visit a travel agent or a website and make one single booking which incorporates your flights and hotel rooms for anything from a week to a month. Once that's booked, there's nothing to do except make sure you've got a valid passport, get the currency and pack your bags. None of those are part of the "booking a holiday" process.
It's a little different if you choose to travel independently. Then you would "book a flight/flights" and then "book a hotel [room]/apartment/B&B/campsite". Even so, once all of those bookings are in place, you can still say "I've finally finished booking my holiday".