Most schools these days seem to offer the vast majority of their places at 11...either for 11+ entry or for deferred entry at 13. Many ask parents to specify one of those entry points when applying. They need to know possible numbers for each entry point when making offers so often aren’t keen to make general offers. If a child didn’t get an offer at 11+ (assuming it’s a school that takes at 11) they often won’t want to take them at 13 either. However, sometimes they will say that applicants can have another go if they do 13+ entry in yr 8 too. The quantity that get taken in via this route varies by school but is declining and declining as schools want more certainty about who is coming. Many will now make more offers for 11 and less for 13 in Yr6. If there are strong applicants they might not take hardly any who test and get offers in yr8. In some areas where there is a strong relationship with certain preps, they might hold more places to be allocated at 13.
So if looking at Preps and senior schools you really need to drill into how many go at 13 and how many have their offers in yr 6 and how many do t get them until yr8. Increasingly it is smaller numbers.
That said, 13+ Preps do take students into yr 7 from outside. They could be coming from Preps who finish at 11 and already have their 13+ offers or they could be coming from Preps or state schools and have not got the 11+ offers they hoped for. Preps which feed a wide range of senior schools and also who lose a good number at 11+ will take such students. However if they haven’t had offers at 11+ they usually caveat it with the fact that parents have to be realistic about which schools should be applied for at 13. If candidates weren’t successful for top schools at 11 they aren’t likely to be successful at 13 in most cases except where there’s been something very unusual.
In the end, all these children from Preps who are there in yr7 and 8 go on somewhere. Preps pride themselves on finding the right school for different pupils and sometimes it’s a case of families adjusting expectations over time. For the more immature boy (and it is the fact that boys more than girls can sometimes not be ready for exams or a move at 11) that extra 2 years or first 2 years in Prep can give them time to mature and be much readier for exams and the move to senior. Another couple of years as big fish in smaller pond really works well for some. But there are smaller and smaller numbers of 13+ schools so I think you really do need to drill down into numbers and also exit numbers and destinations from the Prep and I to the senior schools.
Too many people seem to find they end up in yr7 or 8 in a Prep which is pretty tiny as loads left at 11. Increasingly it can be an odd experience as many have already got their offers and aren’t so focused on preparing for exams and are almost just ‘waiting’ to go to senior school, especially if most start at 11. Things have changed a lot in the last 5-7 years. It has gone from those 13+ Preps having most tested at 13 to them being tested at 11. Lots of senior schools have abandoned the 13+ entry or substantially reduced it. Lots of Preps have abandoned yrs 7 to 8 as numbers have reduced. The Commin Enteance which was the hallmark of yrs 7-8 is used less and less even by top schools who instead rely on their own internal entrance exams at 11 as a better gauge of future ability rather than content heavy Common Entrance.
But of course there are still Preps which have everyone leave at 13 and seniors who don’t take into 13. They might be your best bets and some of the boarding schools and those outside LOndon and the south East might not be so competitive. But even so, most places will still have been offered in Yr6.
In the end they do all go somewhere and most people seem happy and it’s the Prep’s job to sort it out, so moving to Prep for 7 and 8 may well be a good choice. Good luck as we ar equite late in the day but actually lots if 13+ Preps will have places so you should be okay. Just do your research and know what you’re getting into re exit destinations and don’t just jump at the first offer without fully understanding how many already have offers and where children with your DCs profile tend to go.