Say there is a manuscript such as this that isn't in the Public Domain. I have seen one (don't remember where) where it was a manuscript that the museum said they wanted to share online but is "unreleased" and so copyright protected in some way. In that case, and other cases like it, I am wondering if you are allowed to transcribe the picture of the manuscript into say electronic text form, or what the rules are generally around that.
The question contains an incorrect assumption -- the assumption that this manuscript or any ancient manuscript is protected by copyright. As can be seen in this well known chart under US law an unpublished document, such as a manuscript, is protected for only 70 years after the death of the author, or 120 years after the work was created if the death date is not known, or the identity of the author is not know, or the work is of corporate authorship. A work written by a person who lived from circa 950 to circa 1010 is in the public domain under US law, with no room for doubt. The exact term of copyright protection may vary somewhat under the laws of other countries, but in none of them is a document written by a person who died over 1,000 years ago protected by copyright. This work is in the public domain, and no one owns any copyright in it. (A manuscript whose author died after 1948, or that was created after 1898 might well be protected by copyright.)
[The statement in italics above is apparently incorrect. It seems that under UK law such a work may be under copyright if it is unpublished. IMO this is a bizarre result but such seems to be the law.]
The current owners of the physical manuscript can control who has access to it, and on what terms. Thy could use an NDA or other contract to prevent such people from publishing or distributing the text. But if they allow general access to it, they have published it, and anyone may legally republish it as they see fit.
Any person allowed access under a restrictive contract to which s/he has agreed may do whatever that contract permits, and not what it does not.
A modern derivative work, such as an annotated edition, or a translation, would have its own copyright, but this would not protect the original text.
If the original question means that there is an unpublished manuscript recent enough to be under copyright, then its author or the author's heir or the person or entity to whom the copyright was sold or given or left owns the copyright. In that case, no one may make a copy of the manuscript, including a transcription, or of any part of it, unless with permission from the copyright holder, or in such limited ways as is permitted by fair dealing (in the UK and much of Europe) or fair use (in the US). The question does not give enough detail to judge if the partial copy suggested would qualify -- this always depends heavily on the specific facts of the case.
It seems that under UK law some unpublished works are protected which would be in the public domain if they were published. I am not sure how they can be share without being legally "published" and so lose protection.
In the United States, a copyright holder has the exclusive right to make copies of his work and the exclusive right to make derivatives of his work. The general rule is that transcribing a copyrighted work is an infringement.
Your question doesn't give a lot of detail about why the copy is being made or how it might be used, so it's hard to say whether there might be some exception to the rule. If you want to go further, you can take a look at this guide to fair-use analysis.
To answer the headline question: no, you cannot translate a copyrighted work without permission. That is making a derivative work and that is a right restricted to the copyright holder.
You may or may not have a fair dealing defence which I’m not going to analyse here.
Whether a work is subject to copyright depends on the law as it was at the time where it was first published (and the way other nations laws interact with that.
There was no copyright in anything before there was a copyright law in the relevant jurisdiction. The earliest such law was in the United Kingdom and its dominions in 1710. Anything published anywhere before that never had copyright protection. Many things published after that have had copyright protection that has lapsed.
Of course, the physical object is owned by someone and they don’t have to let you copy it if they don’t want to.
A derivative work, such as a photographic reproduction may have its own copyright irrespective of if the original work does. Whether is does depends on if the derivative meets the relatively low standard of creativity required to trigger it. A simple copy doesn’t but if skill beyond mechanical reproduction is involved (as it would be for delicate documents) it probably does.
However, that new copyright only protects the new elements, i.e. the photograph can’t be copied but the original words in it could if they are not subject to original copyright. If the photograph allows you to read the original words of the non-copyright original work, you can copy them.
I suspect BL believe that the legal entity Ælfric of Eynsham (c. 950–c. 1010) may assign the work to a publisher and thus achieve copyright; but in fact it is worse, Ælfric possesses copyright until 2039 ( https://www.thebrandprotectionblog.com/the-2039-rule-uk-keeps-millions-of-very-old-unpublished-works-under-copyright-protection-for-another-24-years/ ).
Their suggestion that it is PD outside UK, and that scholarly uses are unlikely to cause an offence are probably useful. ( https://www.bl.uk/collection-items/~/link.aspx?_id=ECD55B95AD4C4306809EAD127933643C&_z=z )
If you are in the UK find a friend elsewhere who happens to transcribe PD material to PD? Or if you’re an anarchist transcribe, release, and dare the estate of Ælfric to come after you.
I am following the hint in the comment that the question is about converting a picture of a work to text. In which case, you may make a transcription of the work if and only if the underlying work is in the public domain. 1: Assume the underlying work is a manuscript of a book written 100 years ago and still protected by copyright. Whether you purchase a print version, borrow a print version, or see a print version in a museum, you may not photography, xerox, re-type (transcribe), or in any other way make a copy without permission: copying is copying, it does not matter how you copy the work. It also does not matter whether you copy from an officially printed version or a photograph of a xerox of a printed version.
2: Now assume the underlying work is 1000 years old and is no longer protected by copyright. You may transcribe the text as you please, whether this is from the original parchment or a printed photograph of the parchment. You may not photograph or xerox someone else’s photographic version of the original manuscript (assuming that the photograph has the required modicum of creativity – it would, in normal ancient manuscript photography). The photographer has added something to the original work which is protected, and copying the image copies that added protected material. Transcribing does not copy what the photographer has added, it only copies the original text which is no longer protected. (This omits the possibility that you have a separate contractual obligation to not transcribe, which might arise if you got access to the manuscript image though some agreement)