If you pay the credit card off each month, it will be interest free, so you should do that as a matter of routine.
Your cash savings should include an amount that can pay any annual, irregular and unexpected expenses without using credit.
You say you have just remortgaged, if it is a fixed rate, there will be a limit, usually 10% of the outstanding amount, that you can repay without penalty, so you need to bear this in mind.
Also the interest rate on your mortgage, it could be so low that you can get more interest in savings anyway, there have been many deals advertised recently with an interest rate of under 1% and you can get slightly above this rate on the best instant access savings anyway, so it's actually costing you money to overpay, so if that was the case, you could just save the money instead, but bear in mind that if you ever had to claim universal credit, this would be affected by savings above £6k, and you'd not be entitled at all if you had savings of over £16k.