Yes, it is perfectly legal for one nation to “publish” a NOTAM for territories in another nation.
(I changed your wording from “issue” on purpose, I will explain at the end once I have established the basis for it…)
First, let’s define what a NOTAM IS:
From the FAA:
A NOTAM is a notice containing information essential to personnel concerned with flight operations but not known far enough in advance to be publicized by other means. It states the abnormal status of a component of the National Airspace System (NAS) – not the normal status.
More information can be found on the FAA website here: FAA NOTAMS
Another good definition from Theconversation.com:
The NOTAM system is a computer network run by the Federal Aviation Administration that provides real-time updates to crews about situations relating to weather, infrastructure, ground conditions or anything else that may affect the safety of flight.
To summarize, the NOTAM system is means for dissemination of information in a positive and collaborative manner for the purpose of enhancing awareness and safety for flight crews and passengers. It is managed between friendly nations in a cooperative way.
Next, let’s define what a NOTAM is NOT:
An instrument of foreign policy or means to impose political will over a sovereign nation. Nor is it a means for the FAA to impose regulations domestically, that's what the FARs are for.
Finally, let's discuss how it works: (disclaimer – I have never participated in the creation or distribution of NOTAMS, but have been a user of the output for years and have an educated feel for how things work)
- Local airport authorities and ATC have information that is useful.
- They enter the information into a database of some sort.
- The NOTAM system provides the means of distribution.
- Users extract this information as needed.
To expand on your Heathrow UK example, let's say a maintenance worker driving around the airfield notices a pothole that needs fixing on taxiway B. He/she would notify their supervisor who would notify the airport manager. The airport manager would then coordinate with ATC as well as maintenance to decide on the best time to close that taxiway to fix the pothole. Once a decision is made, that information would be uploaded into a database so that the FAA can disseminate it through the NOTAM system. Any crews going to Heathrow during that time would then know what to expect.
For another example touching on "no-fly zones" mentioned in a comment, let's say local ATC decided to issue a TFR (temporary flight restriction) because King Charles scheduled a PR Tea visit to the controllers in the tower. Same thing, this local TFR would be uploaded into a database for the FAA to extract and publish as a NOTAM.
In other words, the information published in NOTAMS is "grassroots", meaning it originates locally and is pushed up to a central location. The FAA knows nothing of detailed ground operations at Heathrow or the King's tea schedule, so they would not, and could not actually create anything. It isn't their jurisdiction, they have no authority.
Getting back to your word "issue"; if you meant simply that they distribute the information then you could say they issue a NOTAM. But they aren't the originators, they simply publish them using the NOTAM system as a conduit, and this is totally legal. It is likely even required in some cases per IACO agreements.
Questions about how nations deal with border disputes is outside the scope of aviation, and is probably a better fit on Politics or Law SE.