Enfield started off making Bikes before Motorcycles, another gun maker similar to BSA (British Small Arms)Are these Enfields "real" or are they badge engineered jobs? I always notice an Enfield fitted with an engine on the rare occasion I see one out on the road, but I can't say I'm familiar with Enfield push bikes though.
Ach, I think you may be rightBirmingham Small Arms I think.
Edit. Yes, for once I'm right. Hang out the bunting.
https://en.wikipedia.org/wiki/Birmingham_Small_Arms_Company
Enfield had stopped making bikes by the time I got into bikes in the 70's. I wonder if Brown Brothers bought the name?