Is a vehicle 'born' when it is first registered? And to expand on this, at what point in the process of modifying a vehicle would it have to be legally classified as a 'new' vehicle?
In the UK, there are two dates that apply, depending on the circumstances - "Date of Manufacture" and "Date of First Registration". This can lead to some confusion with some of the more technical rules, where one requirement might apply to cars built after a certain date, while another might apply to those first registered after a date.
As Bob says, vehicles can sometimes spend some time sitting on a forecourt or in a storage lot and still be sold as 'new' - i.e. unused.
When it comes to modification, we have a points system, whereby certain major components (engine, transmission, bodyshell etc) each count for a certain number of points. If your car has more than 8 points, it is still legally the same car. If you modify it so much that less that 8 points remain original, it has to be re-registered as a special and has to go through a rigourous safety inspection.
[edit in response to the OP's comment below]
As far as I know, these rules applies to anything, but I might be wrong. I believe like-for-like replacements are fine for most parts, although not for the bodyshell (and possibly the engine). Some parts are even more complicated however, for example in the case of engine swaps, the car has to meet the emission standards based on either the date of first registration, or the manufacture date of the replacement engine, whichever is the earlier...
There is information on the Direct.gov.uk website (Direct.gov - registering a radically altered vehicle), and on the VOSA site (VOSA MOT guide). There is also an MOT guide on the independant UKMOT.com website (UKMOT)
The following is written from the perspective of standard practice in the USA:
A vehicle's actual age is defined by its date of manufacture.
The sales-based definition of "new" vs. "not new" is something of a fiction. The dealer can take delivery of a car from the factory, hang onto it for quite some time and still sell it as new if they can find a buyer at that late date. This dealer-defined "new" is usually of primary importance to the customer since that's when the dealer-supported warranty starts. In the state where I live, this also kicks off the count-down to your first safety inspection: new cars aren't required to get an inspection for two years.
It is not immediately clear if it would be illegal for a dealer to hold onto (for example) a 2008 vehicle, keep it pristine in their showroom and still sell it as "new" in 2012. However, I have to think that there would be an immediate civil case.
Finally, where I live, there is no link between modification of the vehicle and its "new" status. The dealer warranty support, however, will take some hits based on the modifications selected.