I believe the Bank of England base rate you reference is similar to the Federal Reserve Federal Funds Rate in the US. This rate, which I’ll refer to as the central bank rate, is the rate that certain banks can borrow from the central bank for an overnight loan. This rate is not a market rate. It is set by a group of central bankers. They can set the rate to any value they please since they have the ability to produce an unlimited amount of funds (i.e. print money).
Interest rate swaps are used to speculate or hedge against a change in interest rates. For example, if I am currently paying interest on a floating rate loan and I believe that rates are going to rise then I can enter into an interest rate swap with another party and swap my floating rate interest payments with fixed rate interest payments. Interest rate swaps are quoted in terms of the rate paid by the fixed interest rate side.
If rates do not change at all during the life of the swap then I do not receive or pay any additional money. If rates rise during the swap period then I will receive the difference between the floating rate benchmark, which is typically based on London interbank offered rate (LIBOR), and the fixed rate. If rates go down I will have to pay the difference.
So back to your question: Do swap rates determine future central bank rates? Maybe but not necessarily. Central bank rates affect market interest rates but market interest rates do not necessarily affect the central bank rate. Banks can borrow money at the central bank rate and loan it out at the market rate pocketing the difference. A larger spread between these two rates will encourage more lending by banks. The central bank does not have to raise the central bank rate even if market interest rates are rising. Remember they can provide an unlimited amount of funds at any given rate. Usually, however, the central bank will have to raise the central bank rate as the market rate rises in order to prevent massive borrowing by the banks which can result in significant inflation.