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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.
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.