Although the current place is not ideal, would it work in the short term for the animals until you do find somewhere more suitable?
If so, I would forget about waiting until the flat is sold and he gets something bigger, that could take a long time as you said and instead just concentrate on getting the animals over on their time scale.
There are still more than three weeks before the 9th of October so you could still make it without changing your vignette.
I think I saw you mention too that you were considering hiring a pet transportation company, maybe they could even board the pets for a couple of nights just to buy a little extra time and you could go on ahead? You may not be happy with this idea, just trying to throw a few more options into the mix. If boarding them is not an option for you, they should still I imagine be able to get you all ready to go within the time you have available leaving you to concentrate on the rest of the move.