Fellow Leicestershire resident here. This is what I do to 1) speak or leave a message and 2) notify a sale or disposal:
1) If you dial the main number it will go work through a series of options and eventually get to a voice prompt that says "If you know the name of the person or department you require please say it now". Ask for "Firearms Licensing" at that point. Note that it may be that being, yesterday, a public holiday that they may have a "grace day" today, Tuesday?
2) As to notification I use their online Leicestershire Police and go to "Apply or Register". You can then download an actual copy of the infirmation you have entered with its reference number. I keep these in the desktop of my laptop in a separate file named as "Police" but you could also print them off an keep them in a ring binder or documents wallet.
Your local police force - online. Report a crime, contact us and other services, plus crime prevention advice, crime news, appeals and statistics.
www.leics.police.uk
You then get a dropdown with six choices, or so, and select the one relevant to firearms and shotguns etc.
The the bottomost one of the choices which enables you to notify the transfer of ownership of a firearm or shotgun. Which when done you have the chance to view and use a go back choice to correct before selecting a submit choice. When that is done the document has its unique reference number created and also the chance to download it.