In most (if not all) English councils with multi-member wards, the question of how to elect multiple councillors at once does not apply in the normal run of events, because only a third of members are up for election in any given year.
There will be three elections in each four-year cycle and each councillor will be elected for four years. For example, Wokingham had elections in 2014, 2015 and 2016, and will again hold elections in 2018. In a multi-member ward, the elections for the different seats will be in different years.
(This is fundamentally similar to the way elections to the US senate work, though the timing and frequency of the elections is different.)
Sometimes, obviously, it doesn't quite work out like this. If a councillor resigns or ward boundaries are changed it can be necessary to elect more than one councillor at the same time. In this case, if N councillors are being elected, each party will offer up N candidates and each voter will get N votes. The N candidates with the highest total vote tally will be elected. Wikipedia calls this Plurality-at-large voting.