You have a very specific idea of your dream house. This is actually waaay more than most people have when beginning a search, so you'll be able to quickly separate the wheat from the chaff. I did a very similar thing about 5 years back and wound up on St. Lucia 2 years ago. If I may attack each of your ticks:
Caribbean beachfront house - Beachfront is a little difficult to come by, but there are measures in place that create ideal beach space. The first 180 yards of land from the high water line are reserved as an area called "the Queen's Chain", meaning no one can own or build within this buffer and all beaches are public. Of course, the land is steep and volcanic creating hundreds of pocket beaches that may only be accessed by water or by crossing private land. ie creating a private beach. Back the to Queen's Chain, the land immediately abutting this buffer is generally reserved for resort development, which puts a hindrance on 'beach front houses' and a high premium on those that exist. Your best bet is land located low on the hills that has an easy walk to the water. These are pretty nicely available as the land at the tops of the hills is considered the prime real estate and is generally the first to go.
Detached, peaceful, idyllic, tranquil, quiet, private paradise- The vast majority of the island is protected Rain Forrest that will never be developed.
Direct short path or garden leading immediately to a quiet or private beach- see my first response, but it will take some good old fashion leg work to find exactly what you're looking for.
Clean or preferably white sand- Depends on which part of the island, it's all very clean, but the color of the sand depends on the volcanic activity it was subjected to when the island was first formed.
Palm trees surround- yes. and Mangroves, Bouganvillia, Sea Grapes, other tropical flora
Wild monkeys on the same island or in the locality are a bonus- an odd request, but yes, they're in the Rain Forrest
English speaking natives, part of the British Commonwealth- an independent nation as of 1979 but still recognize the crown. There is a parliament based/ British style government in place, and the island is English speaking, but there is still a small French Population in some areas and the local Patois is a mix of each of them.
A tax haven of some sort e.g. low/no Capital Gains or Buying Taxes- Laws have been put in place to act as a proper tax shelter, based off of those that have existed for a longer time in the Cayman Islands.
Upcoming island where prices are still low, i.e. not overdeveloped like Bahamas/Barbados- Very new in the development game, prices in the Area of Cap Estate and Rodney Bay are beginning to rise quickly, but still remain around half that of comparable properties in Barbados (100 miles east). The farther away from those areas you go, the quicker the price drops. There are amazing deals in the southern part of the island, which is much less industrialized or developed, which may be what you're looking for in tranquil and private.
Outside the main hurricane belt completely- The last hurricane to hit the island was 1982 and they have never been hit by a cat 5 and only one cat 4 over the past 200 years. The island is south of the traditional path, and far enough east then when one does pass by, it hasn't had sufficient time to build up strength.
Cheap direct flights from London- Virgin flies direct
Good shops- Again, in the north where it's more developed and is continuously being improved on. Rodney is currently building a brand new, modern mall, including shops, condos, and a casino.
Budget: $300,000 - $1,200,000- It's available, again, through old fashioned leg work.
I hope this can help you narrow down your choices a bit, but I'd say St. Lucia is worth looking into.