The key perspective about cross-pavement charging is that, as far as I have been able to determine, there is no law against placing a cable across a pavement. The Highways Act is often cited (Roads Act in Scotland) but the legislation is cited selectively. The text does say that you cannot place a cable (or rope or similar) across a pavement. But it goes on to qualify the statement … UNLESS potential risks are assessed & mitigated, and users of the pavement are given warning.
In fact, pavements are not mentioned - the legislation refers to "a road" (but "road" can be defined as including an associated footpath).
The main argument put forward by authorities is that of trip hazard. They suggest that ANYTHING placed on a pavement is a hazard, even if a cable protector/guard/ramp is used. Yet other regulations define protective measures which must be put in place when excavations take place on a road/pavement (New Roads & Street Works Act). Think of those yellow plates which cover works in progress. If a low-profile cable protector is a trip hazard, so are excavation plates. Conversely, if excavation plates are not trip hazards, neither are low-profile cable protectors.
On the positive side, if an authority objects to an obstruction, they cannot just remove it or impose a penalty - they must give the "offender" notice to remove it, and the period of notice is not less than one month. So, unless you leave your charge cable across a pavement for an extended period, the notice is probably irrelevant.
IMHO, if you charge across a pavement overnight (when, in general, reduced rate EV tariffs apply), putting out a cable after most people are home and bringing it in before they venture out in the morning, it's hard to imagine a local authority officer coming along and taking action.
The key is not to just drape a cable over the pavement. Be responsible. Spend a few pounds on a proper cable protector and make sure any passing pedestrians can see it (including during the hours of darkness).