The videos shared by Mr Trump, who has more than 40 million followers, were initially posted by Jayda Fransen, the deputy leader of Britain First, a group founded by former members of the far-right British National Party (BNP).
Ms Fransen, 31, has been charged in the UK with using "threatening, abusive or insulting words or behaviour" over speeches she made at a rally in Belfast.
Several leading UK politicians have criticised the president for retweeting her posts, as has the Archbishop of Canterbury Justin Welby, who said it was "deeply disturbing" that Mr Trump had "chosen to amplify the voice of far-right extremists".
And it has led to renewed calls for Mr Trump's planned state visit to the UK to be cancelled, although Downing Street said on Wednesday that the invitation still stood.
In hitting out at Mrs May, Mr Trump first tagged the wrong Twitter account, sending his statement to a different user with just six followers. He then deleted the tweet and posted it again, this time directing the message to the UK PM's official account.
It's clear at this point that Donald Trump won't let a perceived slight or criticism go unanswered - even if it's from a supposed friend. Even if it's from the leader of the president's closest international ally.
So shock isn't exactly the right word to describe the reaction to Mr Trump's initially botched attempt to tell Theresa May to, in effect, mind her own business. This is just another example of the US president's self-described "modern-day presidential" use of social media, where Twitter is a cudgel for score-settling no matter the diplomatic cost.
This video was uploaded to YouTube in 2013. The man in the clip says: "No-one but Allah will be worshipped in the land of the Levant," which could place him in Syria.
The third video originates from the riots that took place in Egypt in 2013, and shows a man being pushed from the top of a building in Alexandria. In 2015, those involved in the the incident were prosecuted, and one man was executed.
White House spokeswoman Sarah Sanders said on Wednesday that Mrs May and other world leaders knew that "these are real threats that we have to talk about".
"Whether it's a real video, the threat is real," she said.
What other reaction has there been?
Mr Trump's actions on Wednesday were criticised by both Democrats and Republicans.
Republican Senator John McCain said he was "surprised" at the president's tweets.
Meanwhile, Senator Orrin Hatch of Utah said that Mrs May was "one of the great world leaders", adding that he had "incredible love and respect for her".
“PM @theresa_may is one of the great world leaders, I have incredible love and respect for her and for the way she leads the United Kingdom, especially in the face of turbulence.” -OGH