I think the OP means that his passport is in the passport office in Dublin. They are not issued in London, best as I know, but rather forwarded to Dublin for processing.
The OP can leave the UK, no problem on a non-EU passport, but may encounter some awkward questions if transiting - like why he has no entry stamp for the UK etc. The more important question is how to re-enter Europe.
In a true emergency, the easiest thing to to travel to Dublin and get an expedited passport - in 24 hours, if the emergency is true, verifiable and the travel is imminent.