Hi,
Welcome to the Forum.
The information you've gleaned so far appears to be correct.

Henry H. Aston - Birmingham - 1863
To my knowledge, Henry Hyde Aston was not in the silverware business for very long, he was primarily a manufacturer of gold rings. He acquired his father's (Thomas Aston) business in the early 1860's and this is likely the origin of silverwares bearing Henry's mark.

Thomas Aston & Son - Birmingham - 1858
The '& Son' was not Henry, but William, Henry's half-brother. Thomas and William were registered with the BAO with the mark 'A&S'.

Henry's business was to continue for many years:

H. Aston Ltd. - Birmingham - 1932
H. Aston Ltd. was acquired by Padgett and Braham in 1969.
Trev.