Notice period question

[Kazziepooes]

Apologies for a seemingly stupid question...
I am required to give 3 months notice to my employer, which I did on 1-Oct. Under UK employment law I’m aware the notice period starts the day after you resign. Therefore, I think my final day of employment should be 1-Jan, however my employer are insisting it is 31-Dec to avoid paying me for the bank holiday. Can anyone help here?
(Months is not a defined term in the contract)

Under UK employment law I’m aware the notice period starts the day after you resign.

There is no such law. 99.999999999999999% of UK employers would have 31 Dec as your last day.

Your notice period usually starts the day after you hand it in, but not always, as fair as I know. @flowery May be able to correct me, though!

In my life (FD) we have always taken the day notice is handed in as the first day, however that is all private sector, and maybe public sector is different

At my place of work it would be 31st December.

public sector, same.
might have been doing it wrong I guess; what legislation?