It's weirdly quite a sweet problem to have in that your dog is just TOO happy! So you can take from that that you are doing something right!
I have known dogs who can wag quite noisy while standing next to something reverberative, but still I would be surprised if it was loud enough to wake someone inside a house, when the dog is outside.....
Are they nice neighbours? I think my willingness to inconvenience myself and my family to meet their needs, would in part be influenced by how willing they have been/I suspect would be to do the same.....
Because you have paid for your garden and you have the right to enjoy it in any legal way, which includes using it to allow your dog to exercise and, yes, be happy!
Those saying, yes, of course you must immediately accommodate them - would you think the same WHATEVER time they said, and WHATEVER noise they objected to? What if they didn't like OP talking quietly in her garden at 5 in the afternoon? Where is the cut off?
(OP, Mumsnet is notoriously anti-dog, and I suspect some of the shittier posters on this thread would have agreed with your neighbours or they had complained about your dog breathing too loudly, and called for his immediate execution.)
There is a lot to weigh up here:
A) how much do I honestly think it's inconveniencing them
B) how mush will it inconvenience me (or my dog) to change my routine
C) to what extent have they earned (or the opposite) my respect and cooperation.
It's a fine balance. Hope it all works out.
And finally, my favourite post was the one that implied you were a shit dog owner because you hadn't yet trained your dog not to wag! The animal branch of the happiness police!!!