Thanks, Amanda.
Lt. W. Matthews, M.M., is listed within a record I have, together with 29 other officers all serving in the 31st Warwickshire (Birmingham) Battalion of the Home Guard. The record lists officers as of 1st February 1941 and so it is very likely that your great-uncle was in from the very beginning (May/June 1940).
I'm glad we have found him. Do you have anything more on his Home Guard service? (I'd be happy to give him a mention on the staffshomeguard website).
Chris