Hi
In this case the spelling difference is of no consequence as I searched all the variations including Thelwell.
There will have been Army barracks in the district but their names would be indexed for the census as normal.
I have searched all the Stockport districts as well but still no Thirlwall's.
There does not seem to be a index for his birth on the Cheshire BMD, so I had a quick look on the
https://www.1837online.com which is an index of country wide BMD and there is a reference to a Philip Thirlwall registered in Brampton 2nd qtr (apr may june) 1838 volume 25 page 30, there seems to be other families in the same district as there were several Thirlwall births within the 3 years that I searched.
Can I suggest that as you know when he married you should obtain a marriage certificate, from that you will gather in his correct age, occupation,his fathers name and occupation as well as maybe a relative as a witness.
I have a chart somewhere(cant find it at the moment maybe someone else reading this can help) that would show with the help of the birth ref. numbers which Brampton it was, there are several but none near Hyde.
http://www.genuki.org.uk/big/eng/civreg/places/b.htmhope you find your family
peterbennett