The musters for the 3rd Middlesex or Royal Westminster Militia are in WO13 in Kew. Not available online. They appear to cover one year each, so should give you where he was each month.
They were Regular Militia so almost certainly served away from their home county. You should also check the Middlesex Record Office as someone may have written a history, which will save you a lot of work at Kew.
His first muster with the 20th Foot should have told you how much bounty he received for transferring to the Regular Army. This to allow him to be sent overseas.
Ken