There's no "waiting time" as such in most council areas because almost all councils now use a choice-based lettings system. You "bid" on (ie, express an interest in) properties and who gets what is decided at an allocations meeting.
The general principle is that those bids in the highest band come first, and when there are lots, the people who've been on that band the longest are the ones that get that property. However, my council aims to get as many people housed as possible in each round of allocations, so they will juggle about with transfer applicants and all sorts to maximise the take-up.
Within the top band, people who are homeless and in temporary accommodation are usually given top priority, followed by people downsizing.
There are also big variations in the size of property most in demand. My council has a surprising number of one-bed properties with no age restriction, but is desperately short of 2-beds. People going for 2 bed properties will usually wait longer than people who need 3 bedrooms. Where my MIL lives it's the other way round: loads of 2-bed flats and houses and a shortage of one and 3-beds.
Make sure you use all 3 bids every fortnight if you can, that will mean you get rehoused much more quickly than if you are picky about location and only bid on one or two places.