Are there any noticeable differences in the English vocabulary between the UK and US editions of J.R.R. Tolkien's works, specifically The Lord of the Rings and The Hobbit?
Like many Tolkien questions, there isn't a simple answer to this.
In the front of new editions of LoTR, there is a very extensive "Note on the text", which goes into some detail on how the text developed over the various editions over the years. The first US edition was an unauthorized one, and when the authorized edition was produced Tolkien produced various corrections and updates to the British text - some of which didn't find their way back into the main version until well after his death. The most famous example is the addition of Estella Bolger (Fatty's sister) to the family trees. At the same time, various spelling corrections were made into the British versions, only some of which were incorporated back into the US version... etc. It's only reasonably recently, with the move to computers, that a single version of the text has been able to be produced.
However, as a short answer to your question, at no point was the vocabulary intentionally adjusted for transatlantic differences.
(Geek note: one of the editors/contributors who helped harmonize the text is Wayne G. Hammond, who was an active participant in the rec.arts.books.tolkien Usenet groups back in the day.)