But if both feet are on the ground you can do just as you please, the law does not differentiate whether you are holding a bike at all (see Basil's link which quotes the case law). There would need to be a specific bylaw against carrying a bike, or parts thereof, for that to be illegal.
In Edinburgh the council have always maintained 'no cycling' signs across the Meadows and Bruntsfield links, but the Land Reform Act specifically gives you the right to ignore them, which nobody seemed to realise until quite recently
In England I suspect you might fall foul of all that medieval land law, but I don't think it will be a traffic offence, just trespass?
* or people’s gardens, farmyards, and land in which crops have been sown or are growing. Plus you can't ride over a golf course (should be obvious) but can walk across.