But Mr Heisenber tells us that it is impossible to measure anything without affecting the measured thing!

All measurements make the circuit "work different" in some way:

Inserting an ammeter to measure current changes the voltage drop along the line;

Applying a voltmeter to measure voltage applies an additional load to the source;

Applying a scope probe can affect the cicuit under test.

The secret is to understand your instruments, and ensure that these effects are always minimal; ie,

The ammeter's impedance is much less than the source impedance;

The voltmeter's load is much less than the test load;

The scope probe's loading is much less than the rest of the circuit;

etc, etc...