Waddingham made the remarks while on the red carpet of the 74th Emmy Awards in Los Angeles on Monday.
It is the first major awards show to take place since the news was announced on Thursday, though little mention of the monarch’s death was made during the live ceremony.
Waddingham was nominated for best supporting actress in a comedy series alongside her Ted Lasso co stars Juno Temple and Sarah Niles, but lost out to US actress Sheryl Lee Ralph, who won for her role in Abbott Elementary.
Speaking to US outlet Variety on the red carpet, Waddingham said: “It makes me emotional to think about it.
“I was upset at how deeply upset I was. I sat with my dad on the sofa and I watched it all night until I couldn’t keep my eyes open because I couldn’t believe it.
“It is like a grandmother dying.”
She continued: “It hit me far worse than I thought it would and the enormous pressure that our king has on his shoulders now.
“He will rise to it immeasurably but I feel for him (with) the weight he has on his shoulders.”
Reference to Charles was also made by British writer Jesse Armstrong as he picked up the final award of the night, outstanding drama series, for Succession.
“It’s a big week for successions, a new king in the UK, this for us,” Armstrong said, as he collected the award.
“Evidently a bit more voting involved in our winning than for Prince Charles.”
“Keep it royalist, keep it royalist,” series star Brian Cox told him.
“I’m not saying that we’re any more legitimate in our position than he is, we’ll leave that to other people,” Armstrong replied, as he went on to thank the show’s production team.
Elsewhere at the awards, UK talent fared reasonably, despite British heavyweights Colin Firth and Jodie Comer losing out on top awards.
Matthew Macfadyen won best supporting actor in a drama series for his role in Succession, saying he was “deeply flattered and thrilled to bits”.
Best supporting actor in a comedy series later went to Waddingham’s Ted Lasso co star Brett Goldstein, who promised not to swear as he accepted the award onstage.
“Last time I was here I was told not to swear and I did, and I’m sorry, but it meant that the feed got cut back home in the UK so my family never got to hear me say this but thank you for this second chance,” he said.
The audio then cut out once again on the feed before Goldstein finished his remarks with expletives, prompting laughter from the audience.
Ted Lasso equalled its record-breaking 2021 Emmy nomination run by picking up a total of 20 nods this year.
Leading man Jason Sudeikis took home outstanding lead actor in a comedy series for his role as the eponymous star, and director MJ Delaney won outstanding directing for a comedy series.
The show later went on to win outstanding comedy series for the second consecutive year.
The annual Emmys show ceremony recognises the best in television excellence, and is organised by the US Television Academy.
Saturday Night Live star Kenan Thompson hosted the 2022 show, which took place at the Microsoft Theatre in Downtown Los Angeles.
The comedian kicked off the show with explosive medley of club remixes of popular television theme tunes including Friends, Game Of Thrones and Stranger Things, reassuring audiences that the tracks were “legally cleared”.
Pop megastar Lizzo later gave an emotional speech on representation, after winning the Emmy for best competition series, for her Amazon show Watch Out For The Big Grrrls.
The singer said she had grown up wanting to see “someone who was fat like me, black like me, beautiful like me” as she accepted the award onstage.
Zendaya won outstanding lead actress in a drama series for her role in Euphoria and thanked the show’s production team for creating a safe space in which to make the “difficult show”.
The actress plays recovering addict Rue in the hit HBO series.
“My greatest wish for Euphoria was that it could help heal people, and I just wanted to say thank you to everyone who has shared their story with me,” she said as she took to the stage to accept her Emmy.
“I want you to know – anyone who has loved a Rue or feels like they are a Rue – I want you to know I’m so grateful for your stories, and I carry them with me, and I carry them with her.”
The Spider-Man star made history by becoming the first black woman to win the Emmy twice in the category, as well as being the youngest two-time winner of any Emmy.