She starts primary at 3.4?
I think the paed is hoping that with specialist pre-school, she will be able to start yr r in ms with her peer group.
To be honest, if you have a school with a language unit, it makes sense to use it - we didn't have one, so dd2 started school with 1-1 support for communication.
You don't actually need a specific dx in this scenario - the fact that the paed recognizes she has sufficient needs to require a specialist placement confirms she does indeed have significant sn (at this stage) of whatever sort.
It's a bit like not needing an actual dx for receipt of dla - you just have to prove need. All a dx does is hang a label on a kid which makes it easier. But you have a list of paed recognized needs, so you don't require a dx.
Fwiw, dd2 waited two years for a dx of cerebral palsy.
and that was with the brain injury and with a full set of therapists from birth, and high rate dla from 6 mos.
It drove me crazy at the time - but in hindsight it was the most sensible thing to do - she had extensive needs that were being met, they just didn't have an official title.
Especially with asd and communication, there are a number of possible dx, especially at this age where development is inno way complete.
There is likely to be only one unit near you, tbh. I would call the lea and ask to speak to the area inco that covers early years for your area. Explain that the paed will be writing to them to specify a language unit, and ask where the Lang unit is (tbh you could probably just look on the lea website). Then you can arrange a visit etc. ask for a transition meeting to be set up. This should be attended by both current nursery staff and 1-1, area inco, slt (and any other involved professionals who can make it) and obviously the new setting. Arrange for dd to visit, with her current 1-1 if possible, ahead of transition.
Waiting for a dx is a pita, but if she is getting the support she needs, (and she is) then they aren't doing it to deny access to services, they are doing it because they need more time to get an accurate clinical picture.
Good luck with the transition x