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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.
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.)