Okay, forget Let it Be. The exact song wasn’t really the point. So let’s generalize the question: Would it be okay to take a popular song back in time and ‘write’ it and sell it before the original artist did? Is it plaigarism to steal ideas that the inventor hasn’t thought up yet? There are two cases that matter, I think. The first is if the original author would have written the song in your parallel universe if you didn’t publish it first. Then you could make a case for theft, I suppose. If there were a time police, they’d probably bust you.
But what about a situation where the original author doesn’t write it? Let’s say that when you go back in time you appear on a street, causing a car to swerve and kill the guy who would have written it. Can you think claim the song for your own?