Hi Stevie,
The subject of copyright in digitized works is still hotly debated, and unfortunately the courts in the UK have not been asked to rule on the matter.
In my view a proper interpretation of the law (both UK and EU law) would hold that anything made by a mechanical copying process (
Youtube video) such as that used to digitize old books and manuscripts, or to microfilm old newspapers, lacks the creative human element to qualify for copyright. If there is human involvement at all, it is generally that they place the source material on the scanner and press the 'go' button, and in terms of copyright law, that isn't sufficient to imbue the resulting copy with something of the personality of the author. Indeed it doesn't even meet the older UK yardstick of 'skill and judgement' by the 'author' of the new work.
On the other side of the argument are the companies who have invested heavily in joint projects to digitize archives and libraries. While there are a number of projects including
GoogleBooks,
Project Guttenberg and the
Hathi Trust, whose motives are altruistic, there are several companies who are looking to monetise their investment, as are many of the archives, libraries and museums with whom they collaborate and whose collections are being digitized. Some of these organisations will claim or imply that their digitized collections are subject to copyright even though the original works are in the public domain, but they are probably wrong in law. However since they can control access to their stocks, they can effectively dictate the terms on which they allow copies to be made. So for instance the
British Newspaper Archive (part of the Findmypast organisation and run in partnership with the
British Library) charges a fee to view online content from their collection. However if you can go in person to either the BL in London or to their storage facility in Boston Spa, you can view the collection for free.
So turning to your specific problem, making transcripts (by which I take it you mean handwritten verbatim copying) of out-of-copyright works is entirely legal. It is also legal to copy the extracts you require by photocopying, photography (eg with a smartphone) or a handheld scanner, but all of these devices may well be banned by the particular institution which holds the item you are interested in.
Libraries, museums and archives are
permitted to make copies of certain items (mainly books and periodicals etc, but can include dramatic and musical works too) on behalf of readers for the purposes of private study even when the works are in copyright, and this includes unpublished works. Readers are required to sign declaration about the use of the copies which are supplied to them. The Regulations have recently been
amended to allow data-mining and other analytical processes to be performed on copyright works for the purpose of private study and non-commercial research. Naturally this pre-supposes the works have first been digitized. Of course none of those specific limitations should apply to works which are no longer in copyright, but in my experience many librarians don't have sufficient knowledge about this aspect and tend to apply the 'usual rules'.
I'm not sure where you found the reference to electronic (ie digital) copies only having a copyright term of 15 or 20 years, but I suspect someone is confusing this with something called '
publication right'. This right comes into being when a previously unpublished work is first legally published. By legally published, I mean with the authorisation of the copyright holder if this is appropriate. Publication right lasts for 25 years from the date of publication and belongs to the publisher. The reason this special right exists is because prior to the 1988 Copyright Designs and Patents Act, unpublished works were protected by a special common law provision that said that the copyright term would not commence until publication occurred. This meant that some works such as old letters and diaries were destined to remain forever unpublished because the author had long since died and it was impossible to trace his/her heirs to get permission to publish. The 1988 Act addressed the problem by saying that all unpublished works would henceforward have a fixed 50 term from 1 August 1989 (when the 1988 Act came into force). This means that works of this type will enter the public domain on 1 January 2040, unless they are lawfully published before then. However none of this would apply to works which had already been published, and had merely been digitized, for the reasons I outlined in my second paragraph.
So to summarise, I suspect that what you want to do is totally unaffected by copyright, and that the problems you face are ones of access, as set down by the various institutions you want to deal with. All of them, including government departments such as the General Record Office, and the county Records Offices, are acutely aware of the need not just to cover their costs, but actually make an operating profit, especially where they can charge fees from those who wish to commercially exploit the assets they hold.