It is a manner of speech. Consider:
'but they have not gathered much support until recently'.
In your sentence the speaker has put the emphasis on 'only recently'.
Had they left sooner they would have caught the train.
If they had left sooner.......
There was a delay which, had they not decided to come, would have saved them a lot of distress.
Because they decided to come the delay caused them a lot of distress.
This is all part of the way we speak.
I think the real reason why"they"comes after" have"in your sentence because after the word"only" you need an inversion.Here is an example:Only when I arrived home did I remember I have forgotten my keys.
I hope you understand
This is called 'inversion'. It means inverted normal word-order (as we do in questions) and it is quite formal. We use inversion after negative or restrictive expressions (never, not only, only, little, no sooner....) at the beginning of the clause. Sometimes, as in the conditional examples above, we use it for emphasis or to make it even more unlikely.
Not only IS SHE beautiful, but she also sings beautifully.
Little DID SHE know what was in store for her
Hardly HAD SHE entered the room when she heard someone screaming in the street.
SHOULD I be mistaken, please let me know.