It looks very likely that the John Jaggers chairmaker and undertaker born c.1780 in Clerkenwell is a child of Richard Jaggers and Elizabeth Spice.
I have an 1873 map which shows the different parishes in the City of London. It has St John Zachary and within this there is a Carey Lane which leads off a Foster Lane. These roads still exist and are just to the North East of St Paul's Cathedral, this may help with finding baptisms (assuming they lived in the same area), I couldn't see a St John's church but the nearest ones were St Anne & St Agnes and St Vedast's.
I hope this helps.