No special branch "clearance" is needed - just a notification.
Send them a fax, keep the acknowledgement of it having been sent, and be prepared for some Plod to phone you in the early hours of the morning with a 3-digit "permission number". The Brits just love their job creation schemes.
The flight itself is a non-event. You have to be VFR, and yes being into Class A it is strictly speaking "Special VFR" and this is the clearance you get from Jersey or whatever. But the actual flight is just like any old VFR flight.