From a May article in The Guardian.
"What happens to my mortgage?
If it’s a fixed-rate mortgage, nothing. And most households are on this type of deal – in recent years around nine in 10 new mortgages have been taken on a fixed rate.
If it is a variable-rate mortgage – a tracker, or a mortgage on or linked to a lender’s standard variable rate – the rate could fall a little if the base rate is cut. But the drop is likely to be limited by terms and conditions. David Hollingworth, of the mortgage brokers London & Country, says trackers sold very recently have in some cases had a “collar” that prevents the lender from having to cut the rate at all. Skipton building society, for example, has a tracker at 1.29 percentage points above the base rate that can only go up.
Older mortgages often have a minimum rate specified in the small print. Nationwide building society, for example, will never reduce the rate it tracks below 0% – so if your mortgage is at base rate plus 1 percentage points, it will never fall below 1%. Santander specifies in some mortgages that the lowest rate it will ever charge is 0.0001%.
You will need to dig out your paperwork to see how low your mortgage rate could go."
As it says at the end, you'll need to check the terms and conditions to see shat what would happen to yours.