There seems to be some confusion between the 1772 Royal Marriages Act and the 2013 Succession to the Crown Act which replaced it.
Approval for marriages under the 1772 Royal Marriages Act had nothing to do with succession rights to the British throne.
Descendants who married without the King or Queen's permission were not disqualified from the British throne. The penalty for contracting a marriage without permission was that the marriage was null and void in British law:
[...]"and that every marriage, or matrimonial contract, of any such descendant, without such consent first had and obtained, shall be null and void, to all intents and purposes whatsoever."
https://www.heraldica.org/faqs/rma1772.html
In other words, even if the couple may have exchanged vows in a wedding ceremony, there was no legal marriage in the eyes of the state.
Naturally, the royal descendant who "married" illegally was not disqualified from anything, since in the eyes of the law they had simply remained unmarried.
But since there was no legal marriage, the "spouse" was officially an unmarried partner, and the children of the relationship were illegitimate, for UK purposes. Their illegitimacy barred them from succeeding to the British throne, British peerages, and (in the old days) British hereditary property.
Therefore, if Ernst August wanted the UK government to recognize his wife as his wife for a visa, for tax benefits, for transferring property, or any other legal business he and his wife might conduct in the UK, he was obliged to request the monarch's permission, according to the 1772 Act.
Descendants who were not in line to the throne were still required to ask permission for marriage under the Royal Marriages Act of 1772. Only one class of descendants was exempted, and that was "the issue of princesses who [married] into foreign families":
"That no descendant of the body of his late majesty King George the Second, male or female, (other than the issue of princesses who have married, or may hereafter marry, into foreign families) shall be capable of contracting matrimony without the previous consent of his Majesty, his heirs, or successors [...]"
The Act was silent on who qualified as a princess married into a foreign family. When legal guidance was sought on the marriage of Princess Louise of Schleswig-Holstein (the younger daughter of Princess Helena) in 1891 and the marriage of Prince Adolphus of Teck (the elder son of Princess Mary Adelaide of Cambridge) in 1894, the Law Officers ruled that they were required to ask permission, as their fathers had been naturalized as British subjects.
https://www.heraldica.org/topics/britain/TNA/HO_45_9841_B10982.htm
https://www.heraldica.org/topics/britain/TNA/HO_44_964_B17152.htm
The Royal Marriages Act did not carve out any exemption for illegitimate children. The reason it did not apply to illegitimate children at the time it was enacted is that illegitimate children were not recognized as descendants in British law, until the family law reforms of the last century.