I live in London and have a DS with autism. I'm sorry to say that all local authorities across the UK are really struggling financially and the SEN support and funding system is pretty much broken everywhere.
That's not to say there aren't good schools doing amazing work - there are.
In order to legally secure support for your DD, you will need something called an EHCP.
This is some general information about them:
www.ipsea.org.uk/pages/category/education-health-and-care-plans
I'm also sorry to say that the wait times for an autism assessment in my LA is two to three years. I think for speech and language support it's at least a year.
However, support at school is legally based on need not diagnosis - as is an EHCP, so you won't have to wait to apply.
As your DD is 5 already, depending on when she's born, you will be looking for either a place in reception class or in Yr1. As you're joining mid-year and she doesn't have an EHCP, it will be a case of ringing up the local authority to find out which schools have places. The local authority only have a statutory duty to find her a place in any school where there's space.
I live in an inner London borough and primary school numbers are actually falling quite fast so you may get lucky.
Every school has a person called a SENCO (this is the special needs education co-ordinator). It will be vital that you speak to this person once your DD is allocated a school place and meet with them to outline your DD's needs before she starts.
If your DD has an EHCP, the way you select a school is completely different.
It would be worth looking at mainstream schools with units attached that support DC with autism. My DS is one of them and it's working really well for him. Each unit is run on different lines though.
There are also specialist schools some run by local authorities, some private/independent.
It's possible to get all sorts of educational settings and therapies named in a good EHCP - that way the local authority will be legally and financially responsible for delivering it.
It's a lot to take in I know!
I don't mean to be negative. My DS is 14 now and is doing far better in school than I could have ever hoped for. This is definitely because he has an EHCP and is in a setting that ‘gets’ him and he has full-time 1 to 2 support.
Do post again if you have any specific questions about the diagnostic or EHCP process as there are some helpful, knowledgeable people on this board.