To provide a little bit of backstory:
My dad’s parents come from the Caribbean (his father from St. Kitts and his mother from the British Virgin Islands), and very little is known about his family history. My grandfather was an orphan and thus didn’t know where his parents or grand parents came from. While my grandmother did know do some older relatives, she passed away almost 6 years ago. I am very interested in my family history and it is really unfortunate that I may never know where my paternal family came from.
I am aware that my ancestors most likely came from Western Africa, but I would really like to have a more specific insight on the specific ethnic group or nation they once belonged to. Does the UK government keep any archives with ledgers that may provide information on where their slaves came from?
Another interesting thing I would like to add is that it seems like my dad’s family has adopted European names. For example, my grandfather’s birth name is Cerdric, while his middle name is VanDyke. Are there any historical reasons behind this as well?
TL;DR: Would like some specific information on the Atlantic Slave Trade.
British slave traders did have preferred regions in West Africa where they purchased slaves. This map is a rough sketch of the various regions in West and Southeastern Africa where slave traders visited. Part of the problem would be, for you purposes, is that the slaves were not exclusively drawn from the coastal regions. One of the most thorough studies on the supply side of slave trading, Miller's Way of Death, has various maps showing how the regions that the slaves were drawn from varied over time and penetrated further inland as the eighteenth and nineteenth centuries progressed (Map 5.1, p. 148). As you can see, from the early sixteenth century until the mid-nineteenth century, the regions were slaves were taken from moved East, South, and North, depending on the time, and which political unit in West Central Africa was trying to capture or purchase slaves. One of the best examples of the likely African background of slaves in British North America is an essay by John Thornton, "African Experience of the '20. and Odd Negroes' Arriving in Virginia in 1619" (WMQ, July, 1998, pp. 421-434). Thorton traces the likely African location of the 25 slaves disembarked into Virginia in August of 1619, but even his work is an educated guess. Without knowing for certain which ship or trader your ancestors came on, it is nearly impossible to know which section of Africa your ancestors were taken from.
The best resource to see the broad strokes of the Atlantic Slave Trade, is the Tran-Atlantic Slave Trade Database, available online here: http://slavevoyages.org/. The trade to the British West Indies exclusively is estimated here. You can see that the major regions in West Africa favored by the British were the Gold Coast (which is made up of modern day Ghana, and surrounding area), and the Bight of Biafra (mainly Nigeria and Cameroon). The British Virgin Islands are not represented in the slave trade database, as the likely carriers of slaves were inter-colonial trading ships. Greg O'Malley estimates that a total of 4,500 slaves were disembarked in the British Virgin Islands by 1700, but I don't believe he says much on the eighteenth century for the British Virgin Islands. His Ph.D. thesis just states that the trade was small, and entirely between the various Caribbean Islands.
St. Kitts is a different beast. We have records of about 560 slave ships that traded directly between Africa and the island. The first voyage dates to 1644, and the last to 1807, when the trade was outlawed. The documented voyages show the transfer of about 120,000 Africans from Africa to St. Kitts. The total estimated number of slaves directly from Africa is about 135,000. There may have been some volume shipped from Barbados or Jamaica, but the former is only significant for the seventeenth century, and the latter's volume was probably equal to the number of slaves exported from St. Kitts. From the slave trade database, the major supplier of St. Kitts was the Bight of Biafra, making up 46% of the slaves from known trading regions (the largest section is listed as "Other Africa", which is largely unknown). The next largest regions are West Central Africa (15%) and the Gold Coast (12%). Again, what makes your task difficult is that St. Kitts imported slaves from every major African region. These rough outlines can give you an idea of where you would want to start searching (Ghana, Nigeria, and Cameroon), but they can only be rough guides to start your research. I can't speak to the naming conventions you mention.