Its a bit difficult to say really, as a buzzing sound can be caused by the spark electrodes not sparking where they should, the drive between the motor and the pump could be breaking down, or the motor itself may have difficulty starting up.....however it is strange that the boiler lights up from cold, so: see the grey coil with 3 wires on the top or side of the oil pump? This operates an internal valve in the oil pump and these break down quite often due to heat.
John
June 2009