airpasty
The 04/09 (today) deadline is for those pilots who require EASA continuity on 01/01/21. You can still apply to IAA after 04/09 and before 31/12 and the application will be processed but the exchange of licences may happen in early 2021.
Austrocontrol have a different approach where the application must be completed by 31/12/20, after which they can then no longer convert a third country UK licence. Last I spoke to them they advised me to look elsewhere as their current demand sees them busy upto 31/12.
Lots of states are honouring applications as long as they are made in 2020. There is no real way of speeding up the process by carefully choosing a quieter state as one of the major bottle necks is the UK CAA.