A Jtype OD will fit into a 1500 powered Midget because the engine is identical to a Spitfire 1500. The OD bolts right up to the 1500 Midget engine with minor modifications (see link, below, with photos of the job). You'll need OD wiring (easy), a shortened propshaft and various small bits that are readily available and not expensive.
J-types are not difficult to find, and you could also use the less expensive D-type OD. You can purchase a used one and rebuild it, or get a professionally rebuilt one (with warranty!) from reliable vendors such as Quantumechanics or Waterloo Drivetrain Specialists for $1,200-$2,000 or so. Google search "Laycock DeNormanville", then come back here and ask opinions about the vendors.
Here's a link to chap's website that shows how he did just this to a 1500 powered Midget recently:
https://www.boicey.com/midget/midget_overdrive.html
The Laycock OD won't fit non-1500cc engined Midget/Sprites without significant modification because the engine (1275/1098 etc.) won't mate it. In fact, I'm sure the Rivergate 5 speed would be far more cost-effective in such cases.
So, if you've got a 1500 Midget, go for the OD. The Jtype OD effectively provides a six-speed gearbox with OD in third and fourth gear. It's also very cool to drive and quite reliable if it's overhauled and maintained properly. Not as reliable as a good Datsun 5 speed, but reliable enough for practically anything except racing/consistent thrashing of the car.
If you can do the work yourself, the OD is less expensive to install, also.
Go for it!