Good question. I had my ILR when I left the UK so I couldnt say for sure what the circumstances for your wife (being on LLTR) would be.
Personally, I didnt inform the Home Office that I was leaving the country because Id already sent in my application for citizenship which I knew would one-up my ILR when approved. So different circumstances all around.
If I had to guess off the top of my head, Id say she probably isnt required to inform the HO considering she's giving up her resident status in the UK willingly and has no intention of returning any time soon. Its not like she wishes to make arrangements for an extended stay abroad with intentions to return as a resident or maintain resident status. However, thats just a guess. Id be very interested if you do find out what the actual requirement or suggestion is so please keep us posted.
Hope you and your wife are well!
PS: Hiya Gary!! LTNS!