Hi there. I am from Durham, so not a million miles away. I do not live there now but can provide some background to the North East in general. I grew up in Durham and moved to London, so I hope I can provide some perspective. I have a child with SEN.
I would perhaps consider Durham or Newcastle - unless there is a strong connection to Middlesborough/Stockton/Teeside.
Middlesborough / Teeside is generally a lot poorer than Durham and is much more industrial / post-industrial. There are a lot of social issues relating to poverty in the urban areas. The wealthier towns around are Sedgefield, Yarm, Middleton etc - but my guess is that the SEN schools are in the urban areas = a lot of driving. Durham is the other way around - the town centre / immediately surrounding areas are the nicer bits, whilst the former coal mining villages are generally less affluent and with fewer facilities.
I know Durham better and can provide details of that. There is a specialist school to the north west of the city (Trinity school) housing in that area of the city (Framwellgate Moor, Pity Me areas) is affordable (compared to London at least). And it is also next to the Hospital (Dryburn, now known as University hospital). The school is within a 5 min drive of the town centre.
Given your child has specialist needs, the biggest hospitals in the North East are in Newcastle - this is a 15 min train from Durham or a 30 min drive.
If you have other children - the best state school in the North East is Durham Johnston - and St. Leonard's is reasonable too (both walktable from Framwellgate moor area)
A house is about £150-200k for a 2bed newish build.
One thing to avoid is traffic. No one takes public transport - so a car is essential unless you are within the city area. The walk from Trinity school to the city centre is about 15 minutes.
I realise I moved your Q to Durham area, but I think unless you have a real connection to Middlesborough area, then you should consider other areas in the North East. I can provide similar info on newcastle if you want.
I can answer most things or get answers from people who might know. My friend is a social worker in East Durham county.