Best places to bike in England and Scotland?

Take a look at the Sustrans site, which has lots of information about safe and pleasant travel by cycle. https://www.sustrans.org.uk/ncn/map/national-cycle-network

This page in particular has ideas for leisure cycling: https://www.sustrans.org.uk/ncn/map/walking-and-cycling-inspiration/great-days-out

unfortunately Sustrans can also be misleading, sometimes dangerously. In my town they have spent huge sums of their publicy funded grant and have nothing to show for it but some signs. This causes two problems for the unwary. Some of the signed routes are safe but go miles out of the way, in one important case along a track which is impassible in winter to avoid a road which is always open. In other places the sign is the only thing they provided and the safeness is no different than it ever was.

Your best is to find a busy bike hire shop on the internet - they only exist at the sort of place you are looking for. You don't say half day from where, if London this will probably involve paths alongside rivers or canals. But elsewhere Forestry Commission land is often a good place to start.

Where are you going to be based? I wouldn't recommend cycling in most UK towns and cities, as the traffic isn't very forgiving to cyclists, unless you are used to urban cycling. Few places have dedicated cycle paths.

Good half day cycling I have undertaken includes Bedgebury Pinetum in Kent and the woods around Symonds Yat on the English/Welsh border. Cycling around the reservoirs at Bewl Water in Kent and around Rutland Water in the Midlands are great half day trips. All are off-road on dedicated cycle paths. There is a cycle path along the seafront around Brighton if you wanted to be on the coast.