Sounds like a right old comms failure here; the issue of a slot time is centralised and automated and should be received by the AO (including it's handling agent, where notified), and to ATC.
Can't see any reason why EGHI ATC are saying "talk to your handler" - they get the slot time and should be using it to (a) plan the departure sequence and (b) ensure slot time compliance.
Extracted from ATFCM Users' Manual:
"Slot Allocation Message (SAM)
A SAM is sent to AOs/ATS any time a flight becomes regulated (new flight entering the system, new period of regulation in the system, in response to an FCM or CHG providing new RVR after a suspension) but at the earliest 2 hours before the last received EOBT."