I did this last year. It took one month to sell the car and I went to the bank with the lady who bought it. When the money was in the a/c I handed over the keys.
My wife is great on the phone, so we knew what we wanted and rang about 20 dealers. Eventually got one an hour and a half away that was promising. We called in later the same day, sat down in front of your man and said, we're after driving this far, have a cheque book here and will quite happily drive the car away today. Whats your best price. He beat our local garage by 700 and we had a new car the same day that we sold the old one.
It was a vw golf and most garages had a couple of different ones ready to go. 3 month waiting lists are for trade ins. They always have a little leeway. We sold the old one for more than we were getting as a trade in and got 1700 or so off a new one.
My advice is to put your car on carzone, wait until it is sold before commiting to another one and then ring every dealer going. They all have pan european warranties now that mean any dealer has to look after it.