The 20-year-old winger has signed a long-term contract for a fee Peterborough described as "sizeable" but undisclosed, completing his paperwork at the club's Bespoke Builds Training Ground just days after featuring on Manchester United's pre-season tour, where he scored in a 5-0 win over Rosenborg.
He has been handed the number 10 shirt, vacated following Abraham Odoh's recent move to Salford City.
Williams joined United's academy from Rochdale in 2019 and won the Under-18 Premier League, the FA Youth Cup and the Premier League Cup with the club's Under-18s during the 2023-24 season, scoring seven goals in 11 league appearances at that level.
He never made a senior appearance for United's first team, instead going out on loan to League Two side Cheltenham Town and, more recently, Scottish Premiership club Falkirk, where he made 22 appearances last season including fixtures against both halves of the Old Firm. Peterborough say they saw off interest from a number of other clubs to complete the signing.
Posh boss Luke Williams welcomed the winger's directness and pace.
"Ethan is a player that will give us another option in the wide areas, he is quick, direct and will be a threat and I am looking forward to working with him," he said.
"He has already got first team experience at a few clubs and joins us after being away with Manchester United on their pre-season tour, so he is fit and ready to go."
Williams said Peterborough's reputation for developing young players had been decisive in his decision to leave Old Trafford.
"The identity of this club is clear, they like to work with younger players and when I heard of the interest, it was a move that I wanted to make. I have been out on-loan at a couple of clubs, have gained a lot of experience of men's football and felt it was the right time to leave United and sign for Peterborough," he said.
"I have enjoyed my time at United and it's always tough to leave a place like that, but there comes a time in your career where you have to make tough decisions to develop yourself and Peterborough is renowned for helping younger players."
Williams could make his Peterborough debut as soon as Saturday, when Luke Williams' side open their League One campaign away at Bradford City, following a Carabao Cup first-round win over Bristol Rovers.