Cummings moved to West Ham as a free agent after coming off contract at Celtic.
The 19-year-old has signed a long-term deal with the Hammers, with the contract length undisclosed.
“It feels amazing to join West Ham United, and it's a dream come true,” said Cummings.
“It’s every young footballer’s dream to play in the Premier League, so when West Ham came calling, it was too good to say no, and I’m looking forward to getting started.
“Since day one, when Mark (Noble) was on the phone, he spoke so highly of everyone here and sold the Club to me.
“I did a lot of research about the Club and there’s a pathway to the first team, and that's all you can ask for as a young player."
Sort of player I love
West Ham football chief Mark Noble also said: "We have been watching his progress for the last year now. He has stood out in Scotland for a few seasons and has already gained a lot of experience.
“He is a young, talented player, and the sort of player I love. He works hard, he's aggressive and he has a big presence.
“We knew he was a talent, and we really believe in him. We hope he can come here and show everyone what he's got because he's got a great pedigree for it."