I don't disagree with the warmer seasons having some benefits. Fish immune systems start shutting down in cold water, under 15.5C. So injury could be a problem. On the other hand the problem bacteria also diminishes in cold water. To rebuild the pond you would have to move the Goldfish into something, so a fish tank inside the house would keep them warm, get their immune system working.
So with that out of the way I'd say anytime of year would be fine. In cold weather some of the plants might be dormant, but they're pretty easy to keep alive anyways in buckets or whatever. Most don't have to actually be in water and it's best to keep them out of water to reduce risk of rot. The Duckweed is an exception of course.
Given your focus on wildlife friendly I can tell you there is an entire hobby devoted to this type of pond call a "Wildlife Pond". Lot's of great info. If you read info about Koi Ponds and Water Gardens you will get very different info, correct for those types of ponds but not for Wildlife Ponds. But of course most people keep something somewhere in between. At any rate, a Wildlife Pond starts up very slowly and goes thru a lot of changes. So because of that I don't think the time of year would matter much. Sooner the better I say.
One small suggestion is to build so the final water level in the pond is above the surrounding ground. You can still slope soil up to the edge of the pond so creatures have no problem coming and going. Just 1" above grade, or even with grade is good enough. In wet clay soil water can get under the liner and cause the liner to float up. When the water level is above grade there's enough pressure so water under the liner just comes up and spills across the ground. A floating liner isn't the end of the world, but can be an unpleasant surprise.
