Who owns "The Pearls"? The legal position is pretty clear and can be found on government web sites.
The key thing to understand is that many different types of protection apply to a roll - the physical roll, the musical performance, the composition, the printed words, the label design. The discussion to date has focussed only on the physical roll and the performance on it.
If the rights for The Pearls are active at all, they will be owned by QRS in America, because mechanical rights are active for everything made 1922 onwards - the copyright term has been extended a number of times by successive legislation. However, many old rolls have not been protected and the rights, even if they can conclusively be demonstrated to be owned by anybody, are no longer enforcable - both Duo-Art and Ampico fall into this category. I suspect QRS would be hard pressed to demonstate active protection of the Vocalstyle catalogue. QRS have a long and honourable record of freely granting recutting rights, even for rolls on the QRS label, so asking them is a matter of courtesy as well as a legal issue.
In Europe mechanical rights are 50 years from publication, so anything made in 1957 or earlier is now in the public domain and can be cut without reference to the original producer or any subsequent owner of the rights. Copyright in the compositions, however, is active in Europe for 75 years from the death of the composer (or the death of the longest survivor for multiple-author pieces).
There is presently a lot of pressure from the UK record industry to extend the mechanical rights term because there are a small number of commercially-valuable recordings coming near to the public domain - notably the Beatles material starting in 5 years' time. The government commissioned a review of the whole area (the Gowers Review of Intellectual Property) which emphatically concluded that there was no public benefit in extending terms. The argument continues!