Probate does not exist in Scots Law. The corresponding process in Scotland is called Confirmation. Some confirmations are also listed in the England and Wales Calendar of Probate if the deceased had property in England. In these cases the full process of probate is not invoked; the English and Welsh courts accept the Confirmation process in the Scottish Courts, and the Confirmation is 'Sealed' in England or Wales. So even though
https://probatesearch.service.gov.uk/#wills is an England and Wales source, there is an outside chance of finding a Scottish Confirmation there.
As for
http://nationalrecordsofscotland.gov.uk/research/guides/wills-and-testaments , wills and testaments in Scotland are indexed online to 1925 only.
After 1925, you can consult the Calendars of Confirmation, which are printed books listing all Confirmations on an annual basis. Confirmation of the will of someone who died in 1937 could take several years, so it is necessary to check several years after 1937.
These books are available in the National Records of Scotland and in other libraries and archives in Scotland. If you are anywhere near London, they will also be in the British Library, because under the Copyright Act of 1911, all publishers were obliged to present a copy of everything they published to the British Library and to other Copyright Libraries - see
http://www.legaldeposit.org.uk/ so you should also be able to consult them in Oxford, Cambridge, Dublin and Aberystwyth.
The Calendar lists the name, date of death, date of Confirmation and amount of the estate, and the name of the person to whom Confirmation was granted. Armed with this information, you can go to the National Records of Scotland and see the actual documents, but you need to order them in advance of your visit, because they are not stored in the same building as the search room and they have to be located and brought in for you.