Why does the number have to be a nonce? If you can find a number that gives a block the correct number of zeros, but it has been previously used to give a totally different block the correct number of zeros, what is wrong with using it?
Finding that a single number solved two blocks would be very similar to finding a SHA-256 collision (Bitcoin uses a double-SHA scheme that I think was designed to address a length-extension concern).
The Bitcoin "nonce" is actually a much bigger number than shown in the video. 32 bits I believe.
That's why current miners tamper with both the coinbase transaction, timestamp AND the nonce to find potentially valid blocks in mining.