Depending on the model : 4/4, PLUS 4, Roadster and the year, the procedure may be different.
My Morgan is a 2005 PLUS 4 with a full metal radiator. The size of the radiator is H/W/T 50*38*9 cm.

A friend of mine owning a 4/4 with a ZETEC take out the radiator through the cowl having only to remove the grill. The dimensions of his radiator are H/W/T 35*34*5 cm.
On my car, the air tunnel is 39 cm width allowing a total gap of 1 cm between the 2 internal wing walls. Impossible to remove the radiator by the front and a said by the underneath despite the removing of the support.

Only solution : remove the bonnet and the cowl. The procedure I applied :
1 – drain the system
2 - remove the bonnet : remove the front bonnet end piece : 2 Allen screws and nuts. Don’t remove the backward bonnet end piece, not necessary and screwed in the wood frame. Pull the 2 bonnets with the bonnet strip together. It is better to do it with the help of an other person (my wife).
3 – remove the registration plate and unscrew the 2 nuts of the plastic plate support, extract it and unscrew the 2 Allen screws.
4 – Take out the grill by pulling it downward, remove the cowl : 4 bolts on each side. The cowl with its riveted rubber seal :

5 – Unscrew the clips of the top hose(2), bottom hose(1) and drain hose(1).
6 – Unscrew the 3 silent blocks of the radiator and remove the 3 hoses from the radiator. The top hose can slip along the metal water pipe and the bottom hose can stay on the pipe going to the motor.
7 – Disconnect the fan and extract the radiator, the critical phase is to arrive to extract the lower bended pipe of the radiator throught the front frame. It is not necessary to disassemble the fan from the radiator.
8 – After refurbishing of the radiator proceed in reverse order. Silent blocks have been changed. Verify that the radiator is not stressed in tighting the screws of the silent block : adjust the brackets and/or use rings in order to compensate the gap. You must not have gap or stress between the upper support of the radiator and the silent block. Nota : when the water is hot, the radiator expands roughly of 1 mm pulling the top silent block which doesn’t need to be compressed.

I finished the work yesterday.