The biggest problem with them being balls from the Campbell-Stokes type sunshine recorders is the size, as the smaller the ball the smaller the card the data is burned onto, so I would think it calls into question the accuracy of any readings if they are too small.
I've found the specs from versions still being made and that gives the size of the sphere as 101.6mm (i.e. exactly 4") diameter, the same as the ones in David's Chance advert, and those offered for sale today say they are made to a British Meteorological Office specification (so in this instance I think size does matter Christine :24:
However, searching through the Science Museum online stuff I found a page here: http://www.ingenious.org.uk/media/4.0_SAC/webimages/1031/1/10311940_3.jpg
which shows a variety of instruments from an exhibition in 1885, one of which might have a smaller ball: see image, second item from left top row. Sadly the writing is too fuzzy to read the caption underneath this item.
It might be worth Neil contacting the NMSI and asking if they have a higher resolution copy of this image and what that second item is... and it'd be also worth asking the Manchester Art Gallery if their balls are attached to those stands or if they are separate.