when can I move my japanese peony

not until the autumn now

and even then, it may not like it.
Peonies hate root disturbance, but it will at least have the winter to recover from the move.

true Llew it will certainly set it back, could you not move the rhododendron? They are shallow rooted and move more easily, though you would still need to wait until autumn to do so.

If you feel you cannot wait, by all means move the plant(s) but take as much soil with the roots as possible and thoroughly water the plant(s) every week until established, usually a year. You could also feed it with a liquid fertiliser to help it settle down too.

Paeonies should be moved in autumn as soon as they go dormant. Plant tree paeonies deeply to encourage more woody stems; herbaceous ones shallow (the resting buds should be above soil level). I moved twelve paeonies the autumn before last and nine of them flowered last year so it can be done successfully

That is good news Andrew - looks like we will have to move one this autumn and I always get twitchy...