The fact you are experiencing them with the main switch off and they are "very mild" suggests it could simple be the result of a TN-C-S supply where your house taps are bonded to the supply "earth" but that is combined on the supply (DNO) side with the main neutral conductor for the network (the 'C' in TN-C-S). Then depending on the load imbalance on the various properties fed from that segment of cable you might see 5-10V difference between the tap (on supply earth) and the ground (true Earth).
Obviously it should be checked in case there is something more serious or strange taking place as from my armchair in Scotland I can't divine the true situation, but the checks you report suggest that might be the case.
If it is the above and essentially "safe" (see * below) but still bothers you there are possible solutions but they are not trivial. One option is if there is the option for installing a
very low resistance earth rod / buried mat to try and pull the DNO earth down and the local Earth potential up to meet it, but that is very hard to achieve if not part of something big, like a lightning conductor network or building foundation rebar that is bonded.
Another solution is to change the house earthing arrangement from TN-C-S (where it uses the supplier's earth) to TT where you have your own earth rod. But to do so also requires some analysis and changes to your CU (consumer unit = fusebox) as then disconnecting on a fault will almost always need RCD protection, and ideally that would be two layers to avoid any single point of failure ( see ** below), so something like a 100mA delay-trip incomer RCD followed by the usual 30mA "instant" RCDs for additional shock protection.
[*] There are cases with the combined protective earth/neutral (PEN) of the DNO breaks, and this is potentially dangerous as the neutral current from everyone on the faulted segment of the supply tries to find its way home by other paths, which can be bonded service pipes, etc. Inside a home the open PEN fault case is not too dangerous (though it can damage equipment due to over as well as under voltage operation, and overheat inadequate bonding conductors) but it is considered a serious risk for some outdoor systems, such as EV
chargers where you might be washing the car at the time so wet, conductive, and on the true Earth. In those cases you must use TT earthing or fancy EV chargers that have open-PEN fault mitigation built in.
[**] The regulations allow for just a single RCD 'layer' so long as it meets the specifications for adequate disconnection. They also recommend that folk test them every 6 months just in case one fails, but hardly anyone does that.
…..at 8.30 on Saturday morning 
