Generally they don't have any training.
But look at Autism education trust they publish reports fairly frequently about upskilling staff and may be working on some kind of accreditation.
ICAN do lots of courses.
PEACH and Ambitious About Autism do courses.
Many teachers will go on a TEACCH course.
NAS run courses including one called Socialeyes which I think is for social skills.
There is something called Achievement for All which is the current new thing for schools to be doing for SN - but thats more a whole school approach.
The Dept of Education advertised a scholarship scheme recently (date expired now but should still be on their website) for courses for TAs working with children with SN. It had a list of eligible courses.
DS has an ABA trained 1:1 and they get ongoing on the job training + an intensive introduction to ABA and they all have degrees. The difference between them and the untrained nursery 1:1 DS had is huge.
SN training is poor, most of the ASD teachers don't have proper qualifications themselves e.g. they will go on the beginners PECS course but not get themselves accredited to teach PECS even though they then go round the LA teaching others and saying they are experts! The level of training for SN is really really poor.
When we were faced with perhaps the tribunal making us use a TA rather than an ABA specialist we asked the ABA consultant how long it would take to teach a TA how to teach social skills and he said a minimum of three weeks and even then some of them would not get it. Given no LA releases their TAs for 3 weeks of training let alone 3 weeks on one area then you can see that the chances of getting someone outside ABA with sufficient skills is minimal. The only exception is someone who has worked with children with ASC for years and skilled themselves up, but these are thin on the ground.
The same is true of social care - many social care teams work with teens and adults on the spectrum and mostly they have only done a 1-2 day course.
At our tribunal the autism outreach teacher told the tribunal she was an expert in ABA (and so no need for us to use ABA team). When asked what training she had had she said she was booked to go on a 3 day course (she had not even done it yet!). The SEN Officer looked completely confused when the ABA team listed their qualifications and experience and said 'but the parents have been doing ABA and they have no qualifications so surely anyone can do it'. She could not get her head around the idea that you would need more than 3 days training to teach a child with autism. Yet in medicine if you suggested a 3 day course was sufficient to start delivering complex medical interventions you would be laughed at.
In our team we have ABA tutors who are graduates and get enrolled into 2 year training programme + ongoing training (DS has 3 tutors and they all work with other children each week too); a supervisor who has done 5 years minimum ABA as a tutor and sees DS 12 hours per month, a consultant who has over 20 years ABA experience and sees DS every 3 months. I realise thats the ideal but it shows you how far short LA provision falls.
There is a difference between setting the programme and implementing it. The person who sets the social skills programme needs to be really experienced - the TA delivering it does not but only if they have regular supervision and training / guidance from someone with lots of experience e.g. 5 years plus working with similar children and proving good outcomes.
You want the TA to have done some training and have experience with similar children but you also want a much more experienced ASC practitioner setting up, demonstrating and evaluating the programme on a regular basis.
I think the idea TAs can go on a course then go back to school and deliver an intervention in total isolation without ongoing support or advice is a model doomed to fail. Sadly that pretty much sums up SN support in the UK.