The outward similarity between the 1880-1900 back action SL of J Blanch & Son and the 1890-19++ back action SL of H&H is very noticeable and on certain models and period of time, they are near identical internally as well. It is very tempting to assume a common supplier and it would be a brave (or ignorant) person who stated categorically one way or another. Some of them certainly share many Scott patents such as the Gas Check and Perkes cocking rods (Perkes patent assigned to Scott) but they also display some mechanisms that never were Scott patents, some of which were still under patent protection, such as the lever forend catch on the Blanch which although it looks like a Scott item is in fact a Hollis patent (from 3 years previous) and the interceptor sear which is thought of as synonymous with Scott and H&H is in fact a patent by Needham and Hinton. The forend catch on the H&H no 3 was either the Deeley catch or the Anson pushrod so that points towards Westley Richards. Blanch always used a Perkes ejector box (similar to a Deeley but easily identifiable as different), H&H used the Deeley box until they swapped over to the AB or Southgate ejector. (interestingly, the over centre ejector was not patented by H&H nor Southgate 1st, it was a Perkes invention!)
I think the truth of it is that both Makers bought from whosoever they thought could turn out the right product, to their chosen spec, of sufficient quality and in the right time scale. And that might change on a monthly, annual or even contractual basis.
Scott would have been happy to supply their proprietary actions to anybody who paid up promptly, hence the bevy of Scott mechanisms combined with AN Other's mechanisms as specified by the 'Maker'.
I think both Makers had MOST of their guns finished in London but even that is a leap of faith!
One of the interesting quirks of the Blanch lockwork which illustrates that similarly looking guns may have come from very different sources: Most Blanch & H&H back actions feature the Needham & Hinton 'Patent Block Safety' mentioned above but I have owned and worked on several otherwise typical back actions that have been built using the 'spear' interceptor sear that was the speciality of the famous John Robertson. His workshops had a hand in actioning, stocking, screwing and finishing innumerable guns sold by the biggest and best names in the London gun trade and he and Thomas Perkes probably had a hand, or at least a finger, in the vast majority of the guns sold in London between 1880's and 1900.
I have no doubt the Hugh's information of Frederick Scott being H&H's supplier is correct for the majority of their No.3's but I wouldn't be surprised if they used others if he couldn't supply their total needs. Blanch may also have used Frederick but they had no need to as there were plenty of people in Birmingham and London who could produce something very similar but subtly different!