I'm loving these ideas!
Of course, Tolkien could have been a Time Lord himself, having predicted the Great Storm in the Notion Club Papers, which is quite spooky!
Inserting a bit of our favourite Time Lord to Tolkien's world could explain a fair few things - what about Tom Bombadil? That ageless enigma residing in the Old Forest with his 'companion' Goldberry? Hmmm, and the Ring has no effect on him, and he was there when it was all created, suggesting He Is Not Of This World...
Does Bombadil have a sonic screwdriver hidden away somewhere? It might also explain the eccentric dress sense...
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Or how about a story involving the Doctor arriving on Middle-earth and thwarting Melkor's evil plans to create Cybermen? Or maybe only slightly thwarting it, which may explain where the steely ones come from?
