Baron West is a title created in the Peerage of England in 1402. The title has been in abeyance since 1554, although it is possible to argue that it has been merged.
The First Lord West was Sir Thomas West, of Oakhanger, Northampton. He married the heiress Joan De la Warr, through whom his second son eventually became Baron De La Warr. The two titles descended in tandem until the death of Thomas West, 6th Lord West and 9th Lord De la Warr.
At this point, the precise legal situation becomes debatable. As Cokayne points out, abeyance is a modern rule that approximates medieval practice, rather than a period doctrine. Applying the modern rule, both titles descend to heirs general, which left them in abeyance between the daughters of Sir Owen West, and this situation persists to the present day with their respective heirs.
In 1572, the heir male and nephew of the co-heirs general was created Baron De la Warr, but not Baron West, by letters patent. Depending on one's view of the law, this can interpreted as either a settlement of the ancient Barony of De la Warr or a genuine new creation that extinguished the old barony. Cokayne argues that there is no reason to assume that this changed the status of the West title, but given the irregular practice used, it is just conceivable that a modern claimant to West would be refused on the basis that the title was merged into De la Warr along with the estates.
Some genealogical sources claim that Sir Thomas West of Hampton Cantilupe (1251–1344) was summoned to Parliament and became Lord West in 1342, but this is not supported by Cokayne.