Bible translations like the NKJV and the ESV are meant to be descendants of the KJV Bible. Why then are these written in American English as opposed to traditional British English?
And they sung a new song, saying, Thou art worthy to take the book, and to open the seals thereof: for thou wast slain, and hast redeemed us to God by thy blood out of every kindred, and tongue, and people, and nation; [Revelation 5:9 KJV]
It is perfectly appropriate that each nation should have the word of God in their own tongue, whether distinct language or varied dialect.
So in the USA (and elsewhere) there should be available AmE bibles and in the UK (and elsewhere) there should be available BrE bibles.
I have recently written to the author of the new John Metcalfe Version, on this very subject, disagreeing with them that the name of Jesus (and the title of 'Christ') has been hidden from the English reader by stating it in Greek symbols and I quoted the same above text in my letter.
In the specific case of the New King James version, that translation was a project initiated and carried out in America, and aimed at Americans. As such it makes sense for it to use American spelling.
The King James Only movement was the driving force behind the NKJV, and it is a theological movement which believes the the KJV is the "most accurate" translation of the Bible. This theological movement is nowhere near as popular in other English-speaking countries. The people in the UK who prefer to read the KJV do so mostly because they prefer the aesthetics of the language, and such people are almost all comfortable with the original KJV translation and have no need to "update" it. In fact they would prefer not to.
I’ve got the ESV Study Bible and must confess the language has never struck me as being “American English”. Yes, some words are spelt differently but that is hardly an issue. As for the NKJV, I can’t comment because I don’t have a copy.
The manner in which these translations differ from Bibles printed in England simply reflects how these countries have adapted the English language and spellings. Here are some insights into both translations:
The English Standard Version (ESV) is a revision of the 1971 edition of the Revised Standard Version. The first edition was published in 2001 by Crossway Bibles, a division of Good News Publishers. Under noted theologian J. I. Packer, who served as general editor, the translators sought and received permission from the National Council of Churches to use the 1971 edition of the RSV as the English textual basis for the ESV. Difficult passages were translated using the Masoretic Text of the Hebrew Bible, the Dead Sea Scrolls, and other original manuscripts.
The stated intent of the translators was to produce a readable and accurate translation that stands in the tradition of Bible translations beginning with English religious reformer William Tyndale in 1525–26 and culminating in the King James Version of 1611... they sought to follow a literal word-for-word translation philosophy. To that end, the translators sought as far as possible to capture the precise wording of the original text and the personal style of each Bible writer, while taking into account differences of grammar, syntax, and idiom between current literary English and the original languages. The result is a translation that is more literal than the New International Version, but more fluent and colloquial than the New American Standard Bible. Source: https://www.gotquestions.org/English-Standard-Version-ESV.html
The NKJV was commissioned in 1975 by Thomas Nelson Publishers and took seven years to complete. According to the publishers the object of the exercise was to create a completely new, modern translation of Scripture, yet one that would retain the accuracy, purity and stylistic beauty of the original Authorized Version or King James Version.
However, as has been pointed out in a comment, it moved away from the Textus Receptus.
If people living in America are more comfortable reading the Bible in a language they are familiar with, then that’s fine. It’s the message that counts, after all.