Can't speak to a cozy on an Enfield (don't have a cozy.... yet; but I did ride one briefly) but I've ridden a few hacks, including Ivan, my Ural. The hack is like sticking a sheet of plywood out to the side. Heavy plywood. Aerodynamics get much more important, for one thing. Top speed will suffer because top speed is primarily a function of horsepower vs drag, and you have the same horsepower and a lot more drag. Acceleration will suffer from the weight, but a cozy is much lighter than, for example, a Ural or Hannigan so it'll only be excruciatingly slow, not agonizingly slow. Mileage will suffer from the drag. Handling gets interesting; acceleration will make it turn right, slowing down will make it turn left. The first right turn, you might need clean shorts. You will get an upper body workout every time you go out. You will be way too busy to hold a coffee cup in one hand (like a helicopter pilot, a sidecar driver has a lot of things going on all at once). In general, a sidecar is a miserable bastard that totally screws up a perfectly fine motorcycle.
Fortunately, a well set-up sidecar is also a hoot to drive, very practical for schlepping things around, attractive to children and, shall we say, more mature ladies, alike. If you think you have a delay factor with the Enfield now, just wait 'till you see what a sidecar does. If I wasn't already married, I would be up to my ears in ladies.
The cozy rocket is a nifty little rig, a little small (tighter than the Ural) and pretty light, and can be a sweet-handling outfit. Just give up those dreams of someday going the speed limit, and settle for spending 20 minutes extra every time you park, hauling grandchildren and ladies around, and generally having a good time.