I reckon, and this is based on no knowledge only the experience of doing it a couple of times, anytime from now until early autumn would be OK.
Any spawning is over by now. It is warm, so any frogs and newts 'evicted' during the cleaning will not be too stressed.
We have cleaned our pond a few times in the summer without, as far as we could tell, any ill effects.
Removing the newts was easy but the frogs were more of a problem. There were a lot of them(we lost count at around 35) and once put out of the pond and into shelter under bushes the delightful slimy little jumpers immediately hopped back into the ever decreasing amount of sludge.
The drawback of doing it now is leaves will fall in later in the year but that is a problem at any time.
If possible it would be useful to put some of sludge from the bottom into another container to keep some of the bugs and smaller life forms so the can be put back in when you refill the pond. It is also good to leave clumps of weed close the edge of the pond for a while to allow insects to crawl back in.