Driving with fog lights on when the conditions are fine is illegal. I know this because I have been pulled for it a few years ago. Don't think it applies to headlights though
This is what the regs say, although not that easy to interpret:
Requirements about the use of headlamps and front fog lamps
25.—(1) Save as provided in paragraph (2), no person shall use, or cause or permit to be used, on a road a vehicle which is fitted with obligatory dipped-beam headlamps unless every such lamp is kept lit–
(a)during the hours of darkness, except on a road which is a restricted road for the purposes of section 81 of the Road Traffic Regulation Act 1984 by virtue of a system of street lighting when it is lit; and
(b)in seriously reduced visibility.
I read this as you must have two headlights and you must only use them at night or in seriously reduced visibility.
A sunny day would not constitute 'seriously reduced visibility', so I would be liable to be pulled over by the long arm of the law.