SID for EDDL would be via CLN or DVR while a SID to EGPH would be via BPK.
Did the cockpit crew not question with ATC why they were given a northern departure? Clearly not....
Since the company (WDL) flies to various destinations from LCY on behalf of CityFlyer (including EDI) it would appear that as far as they were concerned their paperwork and Flight Plan stipulated EDI as their destination. They therefore had no reason to question the SID received as it corresponded to the destination THEY were given by the company.