so much so that some of her children were recorded as Nesbitt?
Have you checked the birth registrations of those children?
The little girl is registered with both surnames and Blake mothers name
The little boy is registered with both surnames and Blake mothers name
All the Armstrong - Blake births were registered in Middlesbrough RD.
But, as I said, there were boundary changes in the mid 1930's, including for Eston (which was taken out)
Tees Valley Indexes have the first three births, and say the records are held at Redcar & Cleveland Register Office.
So presumably John and Florence were not actually living in Middlesbrough itself, as evidenced by their absence from the electoral registers there. Possibly then in in Eston / Grangetown.
For the later births indexed by the GRO under both Armstrong and Nesbitt, the TV indexes have them only as Nesbitt, and say that the records are held at Middlesbrough Register Office.
Suggesting that when Thomas Nesbitt and Florence Blake got together they moved (possibly not very far!) into Middlesbrough, as evidenced by them being on the electoral registers there and as per the 1939 Register.