For a popular school, it is difficult to get a late entry place for yr7 if you haven’t done the 11+. You won’t be grammar qualified, plus there are usually long waiting lists.
Many schools determine a period when they hold the waiting list until. It is often Jan of Yr7 or end of Yr7. Whilst many don’t want to move once settled at another school, even if a Grammar place comes up, some will. So this makes it difficult.
Lots of areas will say that the only entry points are at 11 or 16. So if you didn’t take the 11+ you have missed the boat. If they have an empty place (through someone leaving) they will usually take someone who passed the 11+ in another area and has moved to the area.
Some schools are more flexible and if they have a place will allow someone who is interested to take an assessment to see if they meet the standard required.
Some councils have a 12+ exam and some schools will let a few students take an exam at a late point after yr8 in case places become available.
So, the answer is that it depends. There is no single answer that applies to all schools.
However, the thing that is broadly true is that Grammars are very popular and over subscribed, so getting a place later isn’t easy.
Independent schools are likely to be more flexible. When people approach the school about a place (whether they have just moved to area or been in it a while but want to move school) then the school will often allow the,not sit an assessment or take things like CAT4 results from their previous school, and might offer a place.
It’s a shame you are now wishing you’d entered your child and didn’t go through the standard route, but things do change for families.