Swimming clubs vary enormously in how competitive they are, what age they take children from, whether or not they offer 'learn to swim' or just take children who are already competent.
In our borough there are three clubs, two are very competitive and one much less so. All swimming lessons are now supposed to conform to the ASA's National Plan for Teaching Swimming which is supposed to standardise lessons across the board, so if you're stage 6 for example at one swim school, then you could guarantee that you'd be stage 6 if you moved to another school. In practice however standards still vary enormously.
At my children's club, we take children from 5 upwards and have a waiting list for non-swimmers as we can only take a few children at that stage. Once they've passed Stage 7 with us they can then move onto the competitive squads. Occasionally children are ready to move into the squads at 7, it's more usual for them to be 8-10. They can't compete in external galas until they're 9 but can compete in club champs and internal galas.
It really doesn't matter how fast they are at 7 or 8 as the only thing which matters at that age is having fun and working on good technique - if their technique is good they will swim faster, and if they're having fun they'll want to keep going. The children who are fastest at 8 are highly unlikely to be the fastest at 16 - swimming should be a long-term sport as until all the kids competing have gone through puberty it's impossible to predict which are going to be the best.
Have a look at your local clubs' websites, call the membership secretary and ask to find out a bit more about their requirements