The idea that a couple could get their marriage annulled if they didn’t consummate it did not apply in Regency England, nor to my knowledge, at any time in England.
There was one interesting case about a man who went through the embarrassment of being declared impotent and getting an annulment, then later married and had a child. Reportedly, one of the judges wanted to overturn the annulment, but another said that it was possible that a man could be aroused by one woman and not another. According to Ella Quinn: "This is [case is] listed in one of the comments in Haggard's or Philmore's reports on cases in the Ecclesiastical Courts."
In the case of one of my favorite books/movies, Jane Eyre, the husband couldn't divorce his wife or annul the marriage because the wife was considered insane when she married. Her husband would have to prove she did not know what she was doing at the time she married. As he did not know there was anything wrong when they married (her symptoms did not fully develop until later) there were no grounds for annulment or divorce. The courts of England believed that insane people often had lucid intervals during which they could marry and would have known what they were doing, which means the marriage was valid.
Thank you to Nancy Mayer, April Kilstrom, Ella Quinn, and the members of the Beau Monde Regency RWA group for helping provide information for this post.