Onno I made a deposit on April 1st which was sent to Birdee on same day (changed status to "Deposit at Birdee requested") and then on April 8th it was processed by Birdee (changed status to "Deposit is being processed" and then "Deposit finished"). Not fast but seems like it should generally work.
You can see all status changes when you click on the payment and then in the top right corner there's a history button. And then at least on iOS when you long-press on one of the entries you get further status information for that history entry, for the payments on Easy Investments account it shows different status messages.