Even if Probate was not applied for, there is a chance that "Letters of Administration" had to be issued .. also done by the Probate Office .. these are issued when there is no will registered, but a financial organisation wants proof that they can issue monies held ....
As an example of this .. when my Dad died (over 45 years ago) it was not a surprise .. he/we knew it was happening.. he went through everything and transferred it to Mums name, bank accounts, house rental, rates (in those days), gas accounts, leccy accounts, sold the car (he couldn't drive at the end anyway) etc etc etc .. right down to pre-allocating widows pension rights .... he was VERY particular and thorough ... but deliberately left no will as he had "done" everything and there was nowt else to do .......
After he passed I set forth doing all the bits as required and following his instructions, and everything went exactly as he had said ..... until .... his work conatcted me ... he was due "superannuation payments" he knew nothing (or had forgotton) about ... the only way of getting that money to Mum legally was to get "Letters of Administration" in lieu of Probate.... took about 6 weeks in all.... once I learned what they were and how to get them !!
I wasn't sure if copies of those letters are kept like copies of probate are .. so "google being my friend" I did some searching ..... and found this information which may be of help to you .. something I was not aware of ... there is also a bit about Scotland that may be relevant...
https://www.nationalarchives.gov.uk/help-with-your-research/research-guides/wills-or-administrations-after-1858/"The Principal Probate Registry was established on 12 January 1858 and keeps a copy of every will proved in England or Wales after 1858, as well as copies of letters of administration.
An application for administration could be made when a person died without leaving a will."Don't know if that is of any use .. but good luck !!