It makes a massive massive difference
With low interest rates, the difference isn't that great any more. You can get near to the OPs rate on various notice accounts so the 'massive massive difference' is something like about £40 a year - based on £8000 in a fixed rate savings account paying 1.7% pa.
And that's a price worth paying if the OP doesn't have much in the way of other savings, so in a lot of cases, there isn't much of a rush to overpay as there has been previously, even if your rate is comparatively high in the current circumstances, which 2.2% is, I've seen rates for new mortgages below 1% these days.
As for 'preparing for a new mortgage' it doesn't really matter whether you overpay, or save the money, it's available to go towards the new property from either source.
Although another consideration would be if you're ever required to claim universal credit, which could be necessary, given the expected recession etc. If you overpay your mortgage, you won't have savings so will be eligible. If you keep the money in savings, it could reduce your entitlement to UC if you need it.
In short, make sure you have some liquid cash to pay 3-6 months living expenses should you lose your income for whatever reason. Make sure it's in an account where you will get at least some interest. Anything above that, overpay the mortgage if you have one.