Perhaps easier if we take an example, an AIB cheque lodged at Postbank on a Monday, at the earliest will be processed by BOI (As Postbank are not a clearing bank, they use BOI for paper processing) on the Monday night, sent to AIB on Tuesday who will update their records on a Tuesday night.
If the cheque is to be bounced, it will appear on AIB out of order reports on the Wednesday and whereas most such cheques will bounce on the Wednesday, under interbank rules they have until close of business on the Thursday to bounce the item. The physical cheque has to be sent back to BOI firstly (Because that is where it came from) and from there onto An Post who more then likely will not get it until the following Monday and the earliest it would be debited from the An Post account is overnight on Monday appearing on your statement on Tuesday.
An Post could build in a couple of extra days to cover themselves if there are delays, for example, if the cheque is not processed initially by BOI until the Tuesday night rather then the Monday night or if there are any courier/postal delays in getting the cheque back from the other bank if it bounces
Legally, an AIB cheque lodged at an Post or any other bank has to be sent back to AIB, the banks cannot simply send a data file like an electronic payment.
Some of the banks backdate the interest on cheques lodged so the customer is not of pocket but I can't say if all the banks do it
There is little incentive for the banks to invest in speeding up the process as the cheque is a declining instruement of payment with volumes down 10-15% last year and IPSO wanting the cheque gone by 2016