The BMT is a completely new venture, and has nothing to do with the Royal Tournament which ceased in 1999. In the case of all things ceremonial, nothing is like the 'good old days' when higher commands of all three services would stage various events. BMT did start as an idea by the incumbent GOC London District, but the event management companies that run the likes of the Windsor Horse Show, etc., are brought in stage the actual event.
As for RAF participation, the first BMT was an Army-only affair and (like all of the acts involved) have to be bought and paid for so the event teams try and mix the line-up every year for obvious reasons. The Royal Tournament of course saw a lead service every year - this is not the case with BMT and relies heavily on London-based units as they don't need overnight accommodation!
Go trawling Youtube for real spectacles from the past - the bi-annual Wembley Military Musical Pageant (last one was in 1985) was 'Army only' (except in 1977 for the Silver Jubilee) but still mustered 1,300 musicians, including one bit where about 1,000 of them (i.e. sans pipes and drums and Light Divison bands) were on the march. You won't see that again!