But couldn't Germany replace the king as well? Maybe with the abdicated Edward VIII, who was known to be sympathetic to them.
Replacing a king is much harder than replacing a prime minister, especially if the old king is still alive and disagrees.
Public officials can be easily replaced against their will – the system is designed for that. A king, on the other hand, remains a king until they die or abdicate. When that happens, the throne passes on according to an established line of succession. Any other king installed by a conqueror would be an illegitimate pretender. The conqueror could in principle force the parliament to alter the line of succession. But because the UK is a common law state without a written constitution, it's far from obvious if the parliament can actually do that under such circumstances without consent from the monarch and their heirs.
And even in England, in A Man Called Intrepid, William Stevenson notes that it was important that the secret services swore their oath to the king rather than the prime minister. That way, if Germany had conquered England and replaced the prime minister, the secret services could have continued fighting without feeling like traitors.