It will be sent on a collection basis to the other bank in England via an agency bank over there. In effect, PTSB will post the cheque off the England, wait for someone over there to check that funds are in place, no reason to stop the item etc, then wait for the English bank to come back to them to say all is ok before funds are released. It's all manual but probably because there are so few sterling cheques being lodged it is not worth any of the Irish banks effort to invest in a more automated process
as an aside, there are plans in the UK to speed up cheque clearing from 2018 onwards with next day clearance of items and this will include the Northern Ireland banks. In effect, if you have a £ cheque drawn on an English bank and lodge it at your bank on a Monday, you will have cleared funds the next day