When can I transplant a young Japanese Maple?

I would certainly advise against moving now as it has barely had time to establish itself. If you must move again then yes when leaves have fallen and make sure you dig out a root ball at least a metre across. Moving the tree again this autumn is going to set it back so the bigger the root ball the better. Also prepare the ground you will transplant into and add lots of humus.

Adding to that - you will have to water it well for its first year in the new position, too!

Oops thanks Spritzhenry forgot to say that. I know Bulbaholic and I have planted a shrub/tree and the realised we had to move but it is far better to plant in final position as it really sets the plant back.