James Austin Johnson, who plays the US president on the US version of the sketch comedy show, played Mr Trump for his first appearance on the UK broadcast.
The sketch was set at Downing Street and parodied the Ukrainian president’s meeting with the Prime Minister earlier this week.
Mr Burnham, played by George Fouracres, shook hands with Mr Zelensky (Freddie Meredith) as he welcomed him back to No 10.
The Burnham character said: “As a show of support we are accelerating the delivery of British missiles.”
“They are being delivered by Evri, so if you’re not home, they’ll leave them with the country next door,” he added, poking fun at the shipping company.
“Not ideal to leave with neighbour on the other side,” Mr Zelensky (Freddie Meredith) responded.
He then referenced the ongoing tensions between Ukraine and the US: “We need America to restore the Starlink internet. Without it, we are vulnerable to drone strikes.
“Also, it is very difficult to stream Celebrity Traitors.”
“I’m way behind. No spoilers – but I think Richard E Grant is going to go all the way,” Mr Zelensky added, referencing the actor’s early banishment from the BBC reality show.
There were cheers from the live studio audience when Johnson joined the sketch as Mr Trump.
He appeared to take aim at comedian and actor Russell Brand, who is on trial in London accused of sexual offences involving six women, which he denies.
Mr Trump said: “I’m in town for the weekend to see my favourite comedian. He’s performing live at the High Court in the role of defendant.”
He then proposed a “job swap” with Mr Burnham and listed issues in the UK that he would seek to address if he were prime minister, including immigration issues, Manchester City’s financial regulation case and the varieties of M&S Percy Pigs sweets.
He also joked about the fatal tiger attack at Yorkshire Wildlife Park, saying he liked “orange predators” and adding: “Game recognises game. Orange predator solidarity.”
This week’s guest presenter was Happy Valley star James Norton and the musical guest was Birmingham-based band The Streets.
Wicked star Cynthia Erivo will host next week’s episode with musical guest Blossoms.
SNL UK continues on Saturdays at 10pm on Sky and streaming service Now.