The linked Daily Telegraph article about Trump's rejection links to the original claim in Teen Vogue, which says the following:
Trump did not keep the Queen waiting. He was due at 5pm and the motorcade drove into the Quadrangle at 4.59pm.
Teen Vogue in turn linked to marieclaire.com as source, which says:
Her Majesty waited (in public and on live TV, no less) for more than 10 minutes. It's unclear whether the Queen was just early for the meet up or the Trumps were actually late
Marieclaire.com or Teen Vogue's headlines[*] do indeed seem to be imprecise or incorrect, as they don't actually reflect the facts in the respective articles.
Snopes calls the claim "dubious":
Depending upon how one reckons, the Trumps were either just about exactly on time or late by a few seconds in arriving for their meeting with the Queen.
The BBC concludes:
The president arrived and met the Queen on time, according to the official schedule. However, he didn't arrive 15 minutes early at Windsor.
tl;dr: The Queen did wait, but Trump was not late. Some minor online publications used imprecise or just false headlines when reporting on this.
"The Queen Did Not Look Amused to Be Left Waiting for Trump" and "President Donald Trump Was Reportedly Late to Meet With Queen Elizabeth II"