In general, UK copyright law protects printed music for 70 years after the author's death. The same applies in those EU countries that have implemented the EU Copyright Directive. On the surface, this law would protect Medtner's compositions until 31 December 2021.
However, if a work originates outside the European Economic Area and the author is not a national of an EEA state, the duration of copyright is limited to that to which the work is entitled in the country of origin. (This is in accordance with Article 7.8 of the Berne Convention under which an author is normally not entitled to longer protection abroad than at home.) Medtner was naturalized in 1948, but works published before 1 June 1957 only qualify by country of origin. All compositions published in Medtner's lifetime are out of copyright in Russia. Most Medtner works are still in copyright in Europe as those originating in Russia before the nationalisation of the Russian musical press in 1918 were simultaneously published in Germany by Jurgenson, Belaieff or Édition Russe de Musique; this was common practice to secure international copyright for Russian authors. However Opp 33-37 appear to be out of copyright as they were not published in Europe until a year or more after their publication in Russia, well over the 30 day limit required for simultaneous publication.
In the USA, the current 70-year rule does not apply to works created before 1978. Works published before 1923 are in the public domain. This applies to all Medtner's works up to Op 38 plus Op 40. Works published from 1923 have an initial copyright period of 28 years which is extended to 95 years if copyright was renewed during the 28th year. I have no personal knowledge of whether this happened for any of Medtner's works, but copyright appears not to have been renewed for the English translation of The Muse and the Fashion. Dover republished the complete sonatas and Fairy Tales, and Kalmus and Schirmer republished some other works, which may be taken as evidence that copyright was not renewed for those works. IMSLP has treated all the published works apart from Op 58 as definitely or probably out of copyright in the USA.
In Russia, a 70-year rule (extended by 4 years for participants in the 1941-5 war) came into force on 1 September 2006. However works out of copyright on 20 July 2004 under the previous 50-year rule stay out of copyright.
Medtner's works are in the public domain in Canada, Japan, China, South Korea, Australia and New Zealand.
In the UK, copyright in a published edition expires 25 years from the end of the year in which the edition was first published. Typographical copyright does not exist in the USA.
The above is based on my own reading of the relevant legislation. I am not qualified to give a professional opinion on the copyright status of Medtner's works.
In the UK, a new copyright law came into effect on 1 November 2013 that will eventually be harmonised across Europe. For recordings that were in copyright on 1 November 2013, protection now lasts 70 years from the end of the year in which a recording is published or otherwise made available to the public (e.g. by broadcasting); or for unpublished recordings, 50 years from the end of the year in which it was made. Recordings out of copyright under the previous 50-year rule, i.e., recordings published before 1 November 1963, remain out of copyright. For recordings made before 1 June 1957, whether published or not, copyright in the UK lasts 50 years from the end of the year in which the recording was made.
The situation is more complicated in the USA. Recordings are generally protected either for life + 70 years or 95 years after creation, but some recordings made before 1972 will be protected by state law until 2067 regardless of their date of origin. Laws in other countries vary widely.
There has been some debate as to whether digital remastering of old recordings can create new copyright. The consensus of legal opinion is that it cannot.