Secondary school waiting list

If the 4th choice school is an “over my dead body” disaster for you, then you had better accept the private school. If it’s just “not what I would have chosen”, then accept it, (so you have a fall back) and hang on in there with the chance of a waiting list place elsewhere.
If you really don’t want to pay out for private school, you will probably prefer to have the 4th place as a fall back rather than nothing at all.
There is a fair chance of some movement on those waiting lists, but both Wren and St Michael’s are very popular schools (don’t know about St A) - and you’ve given no indication of how well she did in the St M’s test or whether you’ve applied for a church or distance place at Wren and how far away you live. Once the LA has sorted out their waiting lists (usually a couple of weeks after offers are made) you will know where you are in the queue for each school, and will have a better indication. If you’re 1-3, you’ll have a good chance of a place, 4-15 a fair chance, and over 15 less so (my experience only). Less at St M’a I would imagine as it’s a small school and places are hard won.
It’s definitely not worth getting your DD hopeful of a waiting list place anywhere as sometimes it just doesn’t happen - the most important thing is that she doesn’t feel where she’s going is second-rate. Does she have any friends going to the 4th choice? - maybe that can help her feel more positive about things.
Also, definitely consider making an appeal. I don’t wholly approve unless there are a strong set of circumstances - but I have seen many families inexplicably jump waiting list queues this way, just because they they can put together an persuasive argument. It seems very unfair on less articulate families, or those whose sense of fair play allows them to accept that the better / closer candidates won the places - but in this dog-eat-dog North London world, if you can’t beat ‘em, join ‘em.
Good luck - hope this helps.