This scenario can work only on paper. It can’t in practice, neither with a parliamentary system (like the UK) nor with a presidential system (like the US).
In theory, there’s nothing that prevents a prime minister or president to let someone else govern. This other person doesn’t even need to be a vice prime minister, vice president, or anything. Obama could let his dentist decide everything and just sign whatever needs to be signed.
But in practice, in a parliamentary system, the parliament would cast a no confidence vote hence kicking out the government, and in the American system, the president would be impeached for pulling such a trick.
Of course, there might be some countries where there’s no way to remove the head executive from office, but if they exist, I don’t know which ones. And even then, I suspect the legislative body would find a way around (for instance using provisions related to the mental incapacity of the president or something like that).
In fact, in a mature democracy, there’s no way to govern without the assent of the people. Elizabeth II can’t govern all by herself even though technically she could. The French president can’t take dictatorial powers (in normal circumstances), even though he has this option according to the constitution. When all is said and done, if the population at large and the political establishment feels that the political consensus on which the system rests has been broken, the issue will be dealt with, even if it means ignoring what’s actually written in a constitution or interpreting it in extremely creative ways.
By the way, this reminds me of Russia, with the couple Putin/Medvedev, with the latter becoming president just to keep the seat until the former could run again for election. I think it’s relatively close to the concept expressed by the OP. However, those men had also the majority at the Douma (and a majority support in the population), so it’s not really anti-democratic (at least not for this reason).