For morris music, pipe and tabor was traditional until late 19th century. The fiddle came into use around then and one of Cecil Sharp's informants said he couldn't dance to the fiddle when it was introduced to replace the pipe and tabor.

I have, rarely, danced morris to pipe and tabor and can assure you that it is magic, if played well.

The push-pull system was invented in Germany in the early 19th century and the concertina around 1830. The English concertina, made by Wheatstone and later Lachenal, was at first a "posh" parlour instrument. Later the "Anglo-German" system combined the construction of a concertina (buttons pointing out of the ends in roughly symmetrical numbers and pattern on right and left) with the push-pull sequence on the German system and many cheap Anglos were made, which it why the instrument was taken up by Irish traditional musicians and became a folk instrument. Other makers (of all systems) also lowered the prices and the instruments became popular with Salvation Army and similar bands, and in the music halls.

When Cecil sharp discovered morris dancing William Kimber was playing an Anglo concertina, but I don't know how long that instrument had been in use for morris by then, or how many other players used it. According to the concertina FAQ that time (about 1900) seems to have been the peak of the instrument's popularity.

Lots more on concertinas at The Concertina FAQ site.

A search on Google for [button box melodeon] comes up with lots of interesting and relevant links. There's lots of links HERE too.

There have certainly been lots of recordings of melodeon players in the 20th century. It's perhaps less easy to know how much it was used before then, but in any case there were no free reed instruments that we'd recognise before 1800 and very little before 1850.

Anahata