The Local Govt Act 1972 article 84 says the letter goes to the chair as below - once the chair gets the letter there is no going back. The Chair should then notify the Clerk to inform the Principal Authority that there is a vacancy due to a resignation. Once that happens too , the only way back is re election or co option.
"A person elected to any office under this Act ... may at any time resign his office by written notice delivered—
(a)except in a case falling within paragraph (b), (c) or (d) below, to the proper officer of the council;
(b)in the case of a person elected to a corporate office in a London borough, to the proper officer of the borough;
(c)in the case of a parish or community councillor, to the chairman of the parish or community council;
(d)in the case of a chairman of a parish or community council or of a parish meeting, to the council or the meeting, as the case may be;and his resignation shall take effect upon the receipt of the notice by the person or body to whom it is required to be delivered.