In the Church of England, deanery is the name given to a group of parishes which form a district within an archdeaconry (the area which is in the care of an archdeacon), Such a group is put under the care of a dean. The term deanery is also used to refer to the residence of the dean or to the post of dean. In the example given, it is used in the latter sense.
The post of dean of Westminster Abbey in central London, however, is somewhat unusual as the Abbey is a Royal Peculiar, a place of worship that is under the direct control of the British monarch rather than a bishop.
The post of dean of St. Paul's Cathedral, on Ludgate Hill in the City of London, is also a distinctive one, as the cathedral is the seat of the Bishop of London and the principal church of the Diocese of London.
The 'deanery of Westminster or St. Paul's' is, therefore, a reference to two very high profile roles in the Church of England.