There is no regent-ee or similar term. The ascension to the throne of the next in line is automatic and immediate upon the death or abdication of the former monarch. Until that time, the current monarch is and remains the monarch. Ascension of the next in line is then later formalized ceremonially.
Regent is the term used for one who is advising or acting in place of a minor or a temporarily absent or disabled monarch - akin to a vp becoming acting president while the still current president is anesthetized.
So to answer your follow up question - 'As a minor bonus question: is it correct to say "King George III had a regent for the latter part of his reign"?' - Yes. Prince George, later known as King George IV, was in fact a Prince and heir apparent at the time of his regency to his ailing father George III. Due to the fact that he was concurrently regent to his father he was known as Prince Regent, retaining his current title and adding his additional title as Regent.
Also, 'The Tongzhi Emperor had* regents of one form or another for essentially his whole reign; he was a...?' ... He was an Emperor with regents...