As I read the question, the deceased person lived in the UK but the only person qualifying as 'next of kin' lives in St Lucia. My answer, below, is based upon that assumption.
If the testator was normally resident in the England or Wales then the application for a Grant of Representation must (
in most circumstances) be made to the Probate Registry covering the place of residence:
http://www.justice.gov.uk/courts/probate/probate-registries
The Probate Service will prepare an oath, which must be sworn (or affirmed) by the applicant. However it's not necessary for the applicant to attend the Probate Registry in person. They can choose to swear/affirm at any office of a Commissioner for Oaths (which usually means a solicitor), which does not have to be in the UK.
However the Colonial Probates Act 1892, together with the Colonial Probates Act Application Order 1965, provides for the 'resealing' of Letters of Administration granted by courts in certain overseas territories (including St Lucia):
http://www.justice.gov.uk/courts/probate/reseals
So EITHER method of applying for the grant could be used. If help was needed in finding a suitable Commissioner for Oaths in St Lucia, the British High Commission in Castries might be able to assist:
https://www.gov.uk/government/world/organisations/british-high-commission-castries