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This stone comes from the porch of the south wing of nearby Pitzhanger Manor. The 1770 wing was designed by George Dance but the porch, with its naming stone, was a much later addition.

The stone can be seen in place in the 2014 Google Street View. There you see a yellow brick porch added on to the red brick wing. Pitzhanger Manor was subject to a major restoration, 2012–19, during which this porch was removed and replaced with a large window. That must have been when this stone was released into the wild and landed here in Walpole Park.

The Listing entry, written in 1950, is titled "Pitzhanger Manor (Public Library)". We think that explains the addition of the porch with the naming stone. The entry has "South wing of 1770 by George Dance the Younger.... plain, contains dining room and billiard room, both in Adam style."

The house was acquired by Ealing Council in 1900. The extension to the north was removed and the new Lending Library was built in its place and opened in 1902 with a 'Lending Library' naming stone above the door. This block was replaced in 1938–40, by a new, slightly larger building, standing today, much renovated, as Pitzhanger Gallery. The front door can be seen here. The Library moved out in 1984.

None of this answers the question, when did the south wing become the 'Reading Room'?  We'd guess it was in 1900 and that the 'Lending Library' stone (now lost) was in the same style as our 'Reading Room' stone.

Much of our information is from Wikipedia.