In the process of researching this, I found an excellent document (PDF) from Fluke, one of the big names in the tester business, that explains a lot of the safety issues. It even explains the "CE" self-certification mess (though admittedly to their own benefit, since their products have UL or similar testing). A good read, IMHO, for anyone new to electrical work.
I see two very different possibilities:
This is always a concern when messing around with possibly live wires. This is why, for example, some multitesters will include probe covers required for use with CAT III environments in order to minimize the exposed metal area and lower the chance of accidental shorts.
A multitester used properly (voltage) should, under normal circumstances, not cause an actual short, except for a "real" short - e.g., one metal probe hits both hot and neutral. But in the wrong mode strange things can happen.
A ground fault occurs when some current that should be going through the circuit (hot to neutral or hot to hot, depending on the type of circuit) instead goes to ground, possibly through a human being. RCD and GFCI are two different (but related) devices used in different areas to help mitigate this risk. You mentioned UK, which as I understand it typically has an RCD for an entire house (or possibly a few, but not one per circuit like the US).
While a ground fault typically refers to "problem in equipment or wiring", it can also happen with a transfer of current between circuits. That can easily happen in the US if GFCI protection is added to an existing house that has neutral wires connected/shared incorrectly - such problems need to get fixed in order for the GFCI to ever work. But such problems can also happen on a temporary basis if you cross wires in the process of testing. If you are on a whole-house RCD, mixing up the neutrals shouldn't matter, but a small leakage of a hot to ground via testing could trip the RCD.