In connection with Mudie's

 Sometimes the simplest ideas are the best

I guessed that the phrase "in connection with Mudie's" on John Pruddah's library labels might be a standard phrase used by Mudie's franchisees, so I searched Welsh Newspapers online for adverts using the phrase and got just under 700 hits

Some of the adverts are obviously repeats but a quick eyeball survey suggests that it occurred multiple times in multiple newspapers, meaning that there were circulating libraries affiliated with Mudie's in not only the major towns and cities of Wales 

but also in those towns we can identify as nineteenth century holiday resorts such as Llandudno

Woodley's Llandudno

and in Aberystwyth which was then both a university town and a popular holiday resort

Book Depot, Aberystwyth

Using the SLV's subscription to Gale Newsvault to search nineteenth century British newspapers produces quite a few hits including this one from the Stirling Observer in Scotland

No Irish newspapers appear and a search of the Irish Times archive only produces articles about Mudie's, but no advertising, similarly a search for 'circulating library' produces a reasonable number of hits, but again no mention of Mudie's suggesting they did not operate in Ireland.

Equally, in 1886 WH Smith sold their railway bookstall business in Ireland to Eason's. Eason's went on to become the dominant chain of booksellers in Ireland and operated their own circulating libraries.