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There was no comment at all originally from the Queen about them calling their child Lilibet. The media, including the BBC, speculated about whether the Queen would approve or not and Harry jumped in and threatened legal action against anyone who would suggest that the Queen hadn’t given them her blessing. He then applied to the palace to support his narrative, which they refused to do because it wasn’t true.
Harry and Meghan had made up their own ‘truth’ as usual, and no more was heard of legal actions.
Whether the Queen cared or not that the baby was to be called Lilibet isn’t known, what is known is that she wasn’t going to agree to sentiments that she had never said. After enduring all the lies that they had recently told on Oprah, I wouldn’t be surprised if she was at the end of her tether with the pair of them by that stage.