Another possibility is that the mower is not a Barford attachment but a modified Merry Tiller or other make. I don’t recall seeing a rotary mower in any Barford leaflets.
If you want a manual for the ‘motor’ I assume you mean the engine, then you are looking for a Villiers Mk10 engine manual. If you want a manual for the machine it is a Colwood B. Dashwood Engineering were the makers.
Looking at the handlebar and trailer arrangement I would say that is a home made modification. As far as I am aware a trailer was not an option for the Colwood.
There is a trader that gets around the vintage shows who in addition to selling new tools also sells good quality old spanners. He has a very good knowledge of the older stuff. If I see him and can remember to ask I will.
I wonder if it would be worth asking the Tool and Trades History Society (TATHS)