There are three separate conditions of aircraft here as far as I'm concerned.
Firstly, ground-runnable aircraft. These require the least work as systems such as brakes do not need to be operational. Examples include the Me410 and Fw190 held at the RAF St Athan Collection and run up on occasion in the early-1980s. Engines do not need to be restored to airworthy standards.
Secondly, taxable aircraft, such as the Lancaster 'Just Jane' referred to earlier. Brakes and other systems need to be in working order, but they do not need to be to airworthy standards. Not to say that there is anything wrong with the standard of work carried out, just that it does not need to have airworthy, certificated parts used, or the work signed off by a licensed engineer. Having been to a couple of these events at East Kirkby, I can attest to the huge popularity of such taxy runs, even with an airworthy Lanc just 15 minutes down the road at Coningsby. A great way to keep an aircraft 'live' without the attendant costs and risks of flying it.
Finally, there are the 'fast taxy' airframes such as the Victor which inadvertently became airborne a couple of years back. Each to their own, but IMHO the practice of roaring along the runway with a Lightning in reheat is nothing short of barking mad, and it is only a matter of time before it all ends up with a smoking hole at the end of a runway somewhere. (Can you tell I'm not a huge fan of kerosene-burning blowtorches at the best of times?

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Interesting that all 3 are much more common in the supposedly risk-averse, Health & Safely ruled environment of the UK than here in the US.