Technically, two bastards became Queens of England.

When Henry VIII divorced Catherine of Aragon on the grounds that their marriage was illegal, he was, by the legal standards of the time, declaring that they had never been properly married. Thus, any children (ie Princess Mary) were not legal children and so were bastards. Mary was packed off to the countryside, where she stayed with a guardian, and was thoroughly miserable.

Similarly, when Henry executed Anne Boleyn, again, it was considered that Elizabeth was also a bastard.

However, as it happened Henry only had the one son, and he had to consider who would inherit if the worst happened and his son died without heirs. There were distant cousins and cousins-once-removed, but he certainly didn't want his Catholic nephew, James V of Scotland, to inherit (bear in mind that, in the end, James V was killed by an army sent by Henry who felt James had insulted him: that's how much he disliked his nephew). So, he issued a law of succession that put Mary and Elizabeth back into the line, after Edward, essentially legitimising them.

As it happened of course, Edward died young, without children, and so Mary, then Elizabeth, and then finally James V's grandson James VI inherited.

(One point of clarification: I don't actually know when Henry wrote this act of succession. James V might have been dead by then, he was killed 5 years before Henry's death. But having England inherited by a Catholic child living in France and engaged to the King of France would have been even worse.)