The delay meant that a traditional pre-show line up for performers to meet the royal party had to be dropped.
William and Kate, who is pregnant with their third child, graced the red carpet at the London Palladium and swiftly headed to the auditorium for the national anthem, marking the start of the evening.
Despite being four-months pregnant, Kate's baby bump was not showing beneath her sequinned cornflower blue, floor-length Jenny Packham gown.
Singer James Blunt, who was due to perform, had to pull out of the performance as the delayed start meant he would not make a later concert appearance, according to a theatre spokesperson.
Artists including Louis Tomlinson, Paloma Faith and Seal were among the line-up for the performance in front of up to 2,000 people at the annual charity event.
The first act from the cast of the 42nd Street musical got off to a false start when the curtain did not go up.
Host Miranda Hart teased the audience about whether they spotted if anything had gone wrong before reintroducing the act.
The annual event is held in aid of the Entertainment Artistes' Benevolent Fund, of which the Queen is patron.
It dates back to 1912 when King George V and Queen Mary agreed to attend a Royal Command Performance at the Palace Theatre in London in aid of the Variety Artistes' Benevolent Fund and its Brinsworth House home for elderly entertainers.
It became an annual event at the suggestion of George V in 1921, when the King decreed that "the monarch or a senior member of the Royal Family, would attend an annual performance, in aid of Brinsworth House and the Royal Variety Charity, once a year thereafter."