The idea that the BOE sets rates is a bit of a fallacy. It does, but it doesn't set the cost of money, which is what sets how much money (mortgages) cost.
The cost of money is really determined by the lowest risk free rate. To understand what this means think about the following.
The US treasury issues bonds that will give you 4% interest for 20 years. If that is the case, why would anyone bother to lend Mrs Miggins of Accacia road any money for 20 years at 1% when they can buy a US bond that will give them 4% that is less risky (the US is less likely to default on its debt than Mrs Miggins) ?
So in effect the US sets the price of money, and everyone else has to follow. So the US is currently raising rates and the amount of interest it pays on bonds. That means everyone else has to raise rates if they want to borrow money. You'll note that the US, Canada, the UK, the EZ/Europe are all raising rates together. They will all continue to raise rates together, pretty much until the US stops raising rates.
It is of course a lot more complicated than the above - there are some other factors like what happened with the mini budget, but the above really captures the issue. The BOE is just following the US and it has no choice but to really. So if you want to blame someone for why rates are rising you can blame the Fed rather than the BOE.
Why the Fed is raising rates ? Well one things for sure, they aren't doing it because they care one jot about UK mortgage holders, they are doing it because they want to stop US inflation.