I'm on my 4th boat and kept them at Nanny Cay for the most part.
I use Richardson's Rigging all the time for anything related to the pointy-thing going up from the deck
. I've also had good experiences with BVI Marine
Management (right by the travelift). If you bought the boat at Nanny Cay then you probably used BVI Yacht Sales and they will be able to recommend appropriate people for the work you need done - they've been in business there a long time and know their people.
Speaking from experience, I would recommend biting the bullet and slogging the 80NM upwind to St. Martin to get labor and parts-intensive work done to the boat. Everything ends up being about half of the Tortola price
. Also, since this is a new-to-you boat, you will be purchasing
all sorts of items for it, and that, too, is best done in St. Martin.