Is there a tool that can easily detect overloaded or overheating cables? I want to add 2 or 3 extra sockets in a property, but the existing installation is a nightmare. I just want something I can plug into each socket to give me a reading for peace of mind (or not). Is there, for example, a tool that draws the rated current and checks for an acceptable voltage drop over time? For example by drawing the maximum rated current and checking for an acceptable voltage drop over a certain time?
It is not clear (at least, not to me!) what your concern is.
Provided only that the circuit was installed correctly, there should be no risk of the cable becoming overloaded or 'overheating'. Specifically, if the size of the cable (and how it is installed - e.g. whether 'buried in thermal installation') is appropriate for the rating of the MCB protecting it, there is no risk of the cable being o'overloaded' to a dangerous extent, and that would remain true no matter how many additional sockets you added (provided they were added correctly').
Virtually any sockets circuits will have far more 13A outlets for it to be theoretically possible to 'plug in' loads which, between them, represent far too great a load (current) for the cable - but if that situation arose, the MCB would trip before any harm came to the cable. Hence, as above, adding more sockets would not change anything.
Asfar as the cable (and risk of it being overloaded) is concerned, total voltage drop is not really relevant - it's the 'voltage drop per metre of cable' (which is directly related to current) that matters, regardless of the length of cable and hence the magnitude of the voltage drop. If a cable of a certain size 1 metre long is safe when carrying a certain current, then the same cable would also be safe (for the same current) if it were 100 metres long.
If you had reason to doubt that the circuit was originally designed/installed correctly (particularly in relation to cable size), then there could be issues (particularly those relating to 'fault protection') other than 'overloading' to consider - but you would then really need an electrician with the appropriate knowledge and test kit to investigate.
Kind Regards, John