From Allerdale Council site:
Co-Option Procedures at a Council meetingOnly Parish/Town Councillors present at the meeting may nominate, second or vote upon a person to fill the vacancy. At this meeting, members should be informed of the names and addresses of anyone wishing to be considered as a Councillor. Councillors present at the meeting must then decide if they wish to nominate any of the persons named at the meeting or any other persons known to them. A Councillor does not have to nominate any of the persons named. Any Councillor may nominate someone for the vacancy, provided the nomination is seconded, that name may be voted upon.
Voting for the new councillor is by show of hands unless the Council has adopted Standing Orders which allow any other form of voting in Council. If there is only one vacancy, a Councillor may only nominate or second one candidate. The chairman should place the names of those properly nominated into alphabetical order and take a vote. Councillors have only one vote each. The first candidate to receive an absolute majority of those present and voting is declared elected. Should no single candidate receive a majority on the first vote, the person with the lowest number of votes is eliminated, voting takes place on the remainder, one vote per councillor.
This continues until one person receives an absolute majority.
If there is more than one vacancy, a Councillor may nominate one person for each seat. Each Councillor has one vote per seat, i.e. two votes or three votes according to the number of vacancies.
The successful candidate is then declared co-opted to the Council and summoned to attend the next Council Meeting where he/she will sign the Declaration of Acceptance of Office to agree to be bound by the Code of Local Government Conduct.