From the letters to the editor (or equivalent?) of the Lancet (link):

[R]abies was only brought under control after the 1867 Metropolitan Streets Act and the 1897 General Rabies Order, Muzzling Order for the Metropolitan Police District, and Importation of Dogs Order, acts which enforced the shooting of stray dogs, strict muzzling of all pet dogs, and draconian quarantine regulations. The last locally acquired case and death from human rabies was in Wales in 1902, and canine rabies also became extinct after this time.

(conjecture follows)

It's interesting that the myth of "the UK has never been rabies free" prevails; although now it is rabies free makes it much easier to remain rabies free; that letter goes on to say that there was a spike after World War One, when servicemen returned with their dogs – there are now ridiculously stringent anti-rabies laws in the UK, and most (all?) canine infections are picked up in foreign lands (nhs.uk).

If your curious about rabies and its history in general, I highly recommend the book Rabid. http://www.amazon.com/gp/aw/d/0670023736?pc_redir=1400187533&robot_redir=1

It's a bit pop-culturey, but well researched into the history and sociology of the disease.