Some records do exist, although what I found is much later than the introduction of the Act in 1853.
The National Archives catalog has General Register Office: Smallpox Vaccination Returns as part of RG 56. However, the catalog description says:
This series contains smallpox vacination returns for the years 1898-1921 and 1938-1941. These smallpox returns contain statistical information not likely to be found in any such detail elsewhere.
These registers do not contain personal information.
Looking at the individual subseries, many of the descriptions are marked
This record has not been digitised and cannot be downloaded.
The records are listed as open access, but you have to get a reader's ticket to view them, or ask for a quote to get copies made and sent to you.
I searched within this group for Nottingham and Nottinghamshire and got no results. However, one group of 1941 records (RG 56/24) is labeled "Norfolk to Yorkshire" so it isn't safe to assume the name of the locality you seek will be in the catalog description.