What is the right seat height for these?

Most sit between ten and fourteen inches from the floor. The severe drop kept the sitter closer to the heat of the fire. Later Edwardian versions sometimes sit a fraction higher, but anything reaching sixteen inches is just a standard side chair.

I have this low chair that looks like a regular dining chair but without the arms, so how can I tell if it was actually meant for a nursery?

The defining trait is the low seat combined with a tall, supportive back. A dining chair without arms will still have a standard eighteen-inch seat height to match a dining table. Look at the proportions. A true antique nursing chair always looks slightly out of balance. The back will seem unusually high compared to the squat legs. They also tend to have much deeper upholstery than dining chairs, often with heavy buttoning to support the shoulders during long nights.

Can they be reupholstered easily?

Yes, they strip back to the bare beech frame quite simply. A competent upholsterer can remove the old fabric without damaging the show-wood. The coil springs and horsehair layers often survive well and just need retying. Keeping those original internal layers maintains the correct profile and saves on labour.

The back on mine wobbles a bit when you lean against it and I am wondering if this is a common issue with the way they were built back then?

That joint takes the most stress. The tall back acts as a lever against the seat frame every time someone leans back. Over decades, the animal hide glue dries out and the dowels work loose. It isn't a flaw in the original design, just a structural reality for armless chairs from this period. A cabinet maker can dismantle the joint, clean out the old glue, and clamp it tight again. It usually solves the problem without needing new timber.