If this is used a lot by high level ambassadors, and used a lot more regularly than you indicate, they'll never be immersed in the culture of the country they are ambassador to, because they can go home on weekends or at the end of the day. It doesn't sound like most ambassadors will be going back and forth, plus the whole point of being an ambassador is that you LIVE in the foreign country.
We already have real time communication and don't really need to do everything in person. Given that the teleporters are rare and there lots of hoops you must go through to use them, it's safe to say that they might not be used except in the case of an emergency--like a diplomatic crisis that can't be resolved by phone, and even then, is Pakistan really going to invite us over during something like that? We can send a liaison, during a crisis, but they are just going to end up phoning home with updates and reports rather than teleporting back and forth.
Sadly, these machines will actually be a source of strife. I must ask, can they be destroyed or rendered inert? Because paranoid countries will disable, disconnect, collapse or otherwise render the teleports unusable. Because it's a way in for a bomb, an army or a strike force.
The receiving side does not have to confirm the teleportation. (But of course is possible to watch over on both sides).
Which I take to mean that they can't stop the teleportation--in that case, definitely a source of strife and problems...
As a side note though, let's look at some of the other assumptions.
The teleports can't be stolen (or relocated), but can be used for
teleport either people and objects.
The teleports have been in this places forever, but people learned how
to use it only recently (10-20 years). This brings questions how is
possible that the teleports are only just in this cities, we may think
that is caused by some technological reason (so the teleports are in
many places and only in cities they work e.g. they need some amount of
energy). I add this assumption because it makes the most sense IMHO.
You say that the teleporters can't be moved and yet are in all the major cities AND they have been there, perhaps as a relic of a long gone civilization? That doesn't make sense, because there will be many in other places because the last civilization would have had settlements and built things in places that the current one doesn't. As to power, power can be run to almost anywhere. If it's an ancient power source, then maybe civilizations have been built on top of it to take advantage.
Otherwise, were I in charge of the government, I would, for certain, have several "off-books" teleportation facilities not in cities, with full power.
Moreover, some major cities would not have them, because the ancient civilization might not have had settlements in those places--or the location is off--like the teleporter in in New Jersey rather than New York, New York.
So, what would change? Not much except the details. It would be the same geopolitical strife, with this thrown in to complicate things and make it interesting.