Your question is fairly general in nature, but your choice of export market is very specific; I presume that you have a specific reason for choosing that market.
You have raised various points, which are covered below, but I will also raise some other points, separately (due to word count limits)
You don’t need to get permission from any government body to export, although you will have to register with HM Revenue & Customs and get an EORI (Economic Operator Registration Identification) number, which is a unique identification number to identify you when you export).
Some goods are subject to export licencing controls (particularly military / strategic goods, foodstuffs, drugs and articles of cultural value). Your products should not be subject to export licencing requirements, but it is your legal responsibility as an exporter to check; see: https://www.gov.uk/beginners-guide-to-export-controls
Any goods which are exported outside the EU will need to undergo export customs formalities. A good freight forwarder will usually be able to take care of this for you.
Goods may need to have import licences to be imported into the destination country (again, I don’t think this is the case for your goods, going to St Lucia. In addition, you may have to comply with local import regulations covering your products, as the products are presumably applied to the body. You may have to obtain health certificates, certificates of free sale etc. Products may need to be tested, and you may have to comply with local labelling regulations (listing contents, language etc).
Goods will also be subject to import customs clearance when they arrive, and may be subject to import customs duties on arrival. This will usually be paid by the importer, but it is important to clarify this, and you should try to find out what those duties might be, as they will have an impact on the eventual sales price of your product.