I've done 8 weeks in Leeward Islands on a charter (out of Tortola and as far south as Iles des Saintes Guadeloupe and return) and am currently planning 2.5 weeks this June out of Martinique down to northern Grenadines and back. So with 4 weeks you can cover plenty of ground in Caribbean. It really just depends upon what type of cruising and environment you are looking for. Seeing you are out of London, I can't comment on the Med but obviously, the Caribbean is easy travel from there.

If it were me with 4 weeks, I'd probably cruise between St Martin down to Domenica and back, hitting nearly all islands in between. Much also depends on how much you like being on the go vs sitting in one place/island. I'm also assuming a R/T charter, and not one way since those can often be more challenging to arrange.