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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.
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.