Timeline for this cheque is as follows
Lodged in TSB on Dec 22nd and processed by them overnight (Day 1)
Sent out by TSB to AIB on Dec 23rd (day 2) and processed by them
Under the interbank clearing rules, once AIB receive the cheque, they have until the close of business on Day 4 to make a decision as to when to bounce the cheque. On this occassion, because of the Christmas holidays, Day 4 did not occur until Dec 30th. (Note AIB can decide to unpay on day 3 if they see fit)
If AIB decided to unpay the cheque, it would take 2 days for the physical cheque and corresponding electronic to touch down with TSB
Again, because of the New Year holiday, this would not touch down with TSB until Monday Jan 4th at the latest. (day 6) TSB would process the unpaid overnight and that would show in your bank account on the morning of Day 7, ie Jan 5th
Business and personnel cheques go through the exact same process and there is no differentiation between the 2 in most banks. Foreign cheques (and I include NI £ cheques in that) will usually take longer as there is no pan European clearing system, hence those cheques get sent out, often by post, and it can take some time for funds to be sent back and the item cleared
Note, by cleared, I mean cleared for fate. Cleared for interest purposes is different and I know some banks claim that interest is backdated to day 1 or 2, however I've never done a calculation to confirm that.
In essence, if you lodge another banks cheque drawn in € and in ROI into your bank account, it could take up to 7 days for it to be cleared and funds made available