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There is not a single F-16 variant used anywhere in the world that has the full capabilities of their American counterparts ... The same holds for any other advanced fighter platform, including the F-35. This includes nuclear carry capability.

To be clear, I sourced that statement. Relevant bit from the article:

In essence, the nuclear sharing agreement provides for member states of the military alliance without nuclear weapons to partake in planning and training for the use of nuclear weapons by NATO. ... Germany, Belgium, the Netherlands and Italy are all part of the sharing agreement. ... This agreement — American bombs guarded by American soldiers on a German base but flown by crews and planes of Germany's military forces, the Bundeswehr — dates back to the Cold War and NATO's nuclear deterrence strategy 

Italy and Germany have the tornado. Belgium only has F16s which could possibly fulfill that role and will soon have the F35. Same goes for the Netherlands. So they do or will have DCA US made aircraft.
NATO's website:

NATO’s nuclear deterrence posture also relies on the United States’ nuclear weapons forward-deployed in Europe, as well as on the capabilities and infrastructure provided by Allies concerned. A number of NATO member countries contribute a dual-capable aircraft (DCA) capability to the Alliance. These aircraft are central to NATO’s nuclear deterrence mission and are available for nuclear roles at various levels of readiness. In their nuclear role, the aircraft are equipped to carry nuclear bombs in a conflict and personnel are trained accordingly. The United States maintains absolute control and custody of their nuclear weapons forward deployed in Europe, while Allies provide military support for the DCA mission

And wikipedia:

In 2022, after the Russian invasion of Ukraine, Germany announced that it would buy 35 F-35 jets to replace the Tornado in its nuclear sharing role. On 10 June 2013, former Dutch prime minister Ruud Lubbers confirmed the existence of 22 shared nuclear bombs at Volkel Air Base. ... In 2022, after the Russian invasion of Ukraine, reports appeared about the possible inclusion of Poland in the NATO nuclear sharing program.

That last bit is something I didn't know, but given Poland is also acquiring F35s, it makes sense.
To be clear, the UK does not use F-16 fighters. Never has.  It's just an incomplete headline, and should have referenced whatever UK fighter the UK is unwilling to provide to Ukraine. (Probably the Eurofighter Typhoon or its replacement, the F-35B.) >

The F35 is dual capable. IRC the F16 and F15 are too. They can carry nukes.
I can imagine that would be one of the reasons they're hesitant to send stuff like that. Russians see a plane that could be carrying a nuclear bomb. It's flying towards the border. They could come to the wrong conclusion, especially if it's a stealth plane, and they're worried they might lose it. >

Any aircraft sold by the US to foreign governments can be "dumbed down" in capability and many are. That certainly includes the nuclear carry capability. 
Regardless, I think this position has more to do with the difficulty of long-term logistics support than aircraft capability.  While it is possible (and not uncommon) for F-16 units to deploy far from their home bases, long-term support is expensive and complex, and requires a ton of training for maintenance and support engineers. 
When you read about a tank system or anti-aircraft missile system needing lots of training, with a fighter aircraft multiply that need by an order of magnitude. It's not that it can't be done, it's just complex and expensive, far beyond the cost of the aircraft itself. >


Any aircraft sold by the US to foreign governments can be "dumbed down"in capability and many are. That certainly includes the nuclear carrycapability.

In the case of some European allies, they often aren't.
Plenty of NATO members 'without' nukes, have nukes on loan from the US, which they are to be given in the event of a war. It's part of the nuclear sharing agreement. Goes back decades.
We could certainly 'dumb down' Ukrainian jets, but what's no point. They don't have nukes, but how are the Russians to tell from the ground?
The F16 maybe, but the F35 has stealth capability. That's an issue. >

Your first stament is simply incorrect as it concerns advanced fighter aircraft.  There is not a single F-16 variant used anywhere in the world that has the full capabilities of their American counterparts. The same holds for any other advanced fighter platform, including the F-35.  This includes nuclear carry capability.
Now, that’s completely different than other types of nuclear weapon delivery and I have no expertise in any of that.
As to whether or not the Russians know whether a given F-16 has all the capabilities of the US market aircraft:  believe me, they know.  But it doesn’t matter, because it’s not the “fully loaded” capabilities that the Russians are worried about, just the basic air superiority and ground attack capabilities that these aircraft afford.  
Nukes aren’t the issue with the aircraft possibility, just basic force multiplication.