Michael Sandford, 20, allegedly tried to snatch the weapon during a rally at Treasure Island casino in Las Vegas on 18 June.
He is said to have told police he travelled to the Nevada city to kill Mr Trump, the presumptive Republican nominee for the US presidential election, according to court documents.
Sandford, of Dorking, Surrey, appeared at a federal court in Las Vegas and pleaded not guilty to a charge of disrupting government business and official functions and two charges of being an illegal alien in possession of a gun.
He was remanded in custody and is due to stand trial on 22 August.
According to a complaint lodged with the US district court in Nevada, Mr Sandford told a policeman at the rally that he wanted Mr Trump’s autograph before he attempted to seize the officer’s gun.
During a police interview, when asked why he attempted to grab the weapon, Mr Sandford replied: “To shoot and kill Donald Trump,” the complaint alleges.
He had driven to Las Vegas from California on 16 June, and went to a shooting range the next day, where he said he learnt how to use a gun, it is claimed.
Mr Sandford – who had been living in the US for around 18 months – allegedly told investigators he expected to die in the attempt on 18 June, which he had been planning for a year. His mother has voiced fears that her son, who faces up to 30 years in prison, may try to take his own life in a US jail and called for him to be returned to the UK.
Lynne Sandford told the BBC’s Victoria Derbyshire programme: “I would like him to be deported so he can be back in the country and can get psychiatric help and that way he could see the family that still adore him. Jail is no place for him. He needs help, but not prison.”
Mrs Sandford said her son had been diagnosed with Asperger’s syndrome when he was 13 and had previously tried to kill himself.