That bike will be great with a 3sp coaster. I’ve completed several conversions myself. The Sturmey hub is slightly wider than the stock 1sp coaster. Most likely you will need to spread the rear drop outs.
Not too hard to do. Remove the stock wheel, the fender, chain guard, at least one pedal, the seat and strut. Removing the extra parts keeps them from any damage when you spread the drop outs. Lay the frame on its side on a pad or the carpet. Place a small 2X4 on one drop out; lay your pry bar on it. Place another 2X4 on top of the pry bar. Gently pry the two 2X4s apart. Keep checking the fit of the Sturmey hub.
(For those of us without a bike stand)
The Sturmey hub is a wide ratio. The stock gearing on your bike, made for a kid, is probably 46 tooth drive gear and a 20 tooth hub gear. I would recommend a 16 tooth hub gear for road travel. If you have strong legs and want to go fast, use a 15 tooth hub gear. If you want to use it as a mountain bike, keep the 20 tooth. With a smaller gear you’ll need to shorten the chain a couple links.
Don’t forget, the Sturmey hub must have anti-rotation washers. Some diagrams may show them on the inside of the drop outs, but they’ll work just fine on the outside of the drop outs also.