We were in a similar position as well, these are the steps we made.
1) Contact your solicitor to get title deeds of existing house transferred to them.
2) Put your house on the market.
3) Start house hunting.
It is probably a good idea to have your house with one of the big estate agencies. When you come to making bids on the other house, it could be with the same agency and may have a better chance getting your offer accepted.
We were bidding on a house in Febuary and got Sale Agreed because our house was on the market for a week or two beforehand. The other bidder hadn't even got their house on the market and thats what swung it for us.
However, doing it this was around, we felt more pressurised. Even if you go sale agreed before finding somewhere else you can always give back the deposit or explan to your bidders that you need a bit a flexability.