An incoming councillor can't accept the position until it's vacated by the outgoing councillor, otherwise you'll have too many councillors. The legislation is Section 83(4) which states:
"A person elected to the office of chairman of a parish or community council or parish or community councillor shall—
(a) in the case of the chairman, at the meeting at which he is elected;
(b) in the case of a councillor, before or at the first meeting of the parish or community council after his election...
...make in the presence of a member of the council or of the proper officer of the council and deliver to the council a declaration of acceptance of office"