The 18-year-old, who arrived at Adams Park last Monday, has been training with the Chairboys and impressed in a 45-minute appearance against Havent & Waterlooville on Saturday.
Fletcher has penned a one-year deal with the option of a second year.
"It's my dream," he told the club's official website. "I've always wanted to be a professional player and I'm just glad that I've signed a contract and got here.
"I'm over the moon to be at Wycombe. It's such a big club and I've never experienced anything like it."
Manager Gareth Ainsworth added: "He's impressed in training and in the game against Havant and did more than enough to make us want to sign him.
"He's got the raw materials and more. With hard work and coaching he can go on to be a very good player. Other clubs were interested in him, and we're pleased to have secured his signature."