Often, the buffer was never copied. Instead, the video card is told to read from the old back buffer (which is the new front buffer), and the old front buffer (which the video card was drawing from last frame) becomes the new back buffer. This only requires a few instructions, so can easily be done during the monitor's vertical blank.
That is entirely dependant on the hardware. Some machines could, some couldn't, and some could but mucked it up.
The Spectrum couldn't so you either wrote to the live video display or used some method of copying.
The QL could in theory switch video display between two buffers but they put a section of IO/System memory (I can't remember the details) in the middle of one of the buffers so it was useless.
Some systems also had hardware sprites which weren't part of the main video ram, they were overlaid by hardware. You could also do some fun stuff with those such as rewriting them as the scanline progressed down the screen to fudge a lot more hardware sprites.
I'm out of the industry now but one of the major parts of games programming was pushing the hardware way beyond what it was designed to do. Now, as far as I understand, you write for an API.
