So I was surprised to see the travel restrictions for the two counties in Wales, and this time the Welsh government website specifically says that you can't go on holiday, which is a step up from before. Though I suppose Wales has done some things a little differently, for example writing the 1 hour of daily exercise into law, which was not the case in other UK countries. However reading the legislation posted by KARFA it is worded in the same way as the initial restrictions - you need a reasonable excuse to leave but there is only a suggestion of excuses that are definitely considered reasonable and basically leaving it up to the police or a judge to decide.
The two ones in Wales are outliers really as they impose restrictions on people leaving the designated area, and on people from outside entering. The original national lockdown was based on a reasonable excuse to leave your home per se, rather than leaving an area.
None of the local lockdowns in England include anything like the two in Wales. Instead the ones in England merely prevent gathering of people from two or more households in private dwellings unless they are a linked household (commonly known as the social bubble allowed if one household is is a lone adult) or some specific other exemptions. Also the England local restrictions apply to the person whether in the designated area or not. So someone in a local lockdown area can't gather in a private dwelling with another household in the designated area, and additionally can't gather in a private dwelling with another household even if that dwelling is outside the local area. So if you are in the North East you can't visit another houshold in say Cumbria even though there is no lockdown in Cumbria. So you are not limited in your travel, but you are limited in gathering with others in private dwellings anywhere in the country.