The clubs agreed a deal for the Bury academy graduate in November and he became a Toffees player when the transfer window reopened on Saturday.
Foulds signed his first professional contract with League One Bury in February and made two first-team appearances, both in cup competitions.
He has also played on a trial basis for Everton's Under-18 and Under-21 sides.

The PSNI began an investigation after allegations emerged last week that trainees had shared exam questions.
Chief Constable George Hamilton has said 54 student police officers must go back to the start of their 22-week training programme and repeat it.
He said all concerned received written warnings under misconduct procedures.
Several recruits were not allowed to graduate from the PSNI training college at Garnerville, east Belfast, after a whistleblower raised the allegations.
In a statement, Mr Hamilton said the cheating claims emerged "the day before last Friday's graduation".
"The allegations of impropriety suggested that student officers had individually memorised examination questions and collectively shared this information between themselves with a view to assisting them prepare for any re-sit examinations they might have been required to take."
He added: "As chief constable, I am deeply disappointed by the actions of those student officers who have acted in a way that is not in keeping with the standards I expect from aspiring police officers. There have been disciplinary consequences for all of those students who were involved in this impropriety."
However, Mr Hamilton also admitted that Policing Board members, who hold the PSNI to account, were "concerned at the action I had taken".
He described his meeting with board members on Thursday as "challenging".
He added: "I accept that others may not fully agree with my decisions on the matter.
"However, I have acted in good faith and with integrity believing that my actions have been proportionate and appropriate in all the circumstances."
In a statement, the Policing Board said that the chief constable was questioned for two hours regarding the "actions and decisions taken following this information coming to light".
Policing Board Chair Anne Connolly said: "The chief constable has been left in no doubt that board members consider this to be a very serious matter which has caused reputational damage to the PSNI.
"It is deeply concerning to the board that so many trainees, at the very start of their careers, have been prepared to engage in this impropriety."
She added: "Members questioned the chief constable on their suitability and future credibility to perform the duties of constable and did not agree that the sanctions imposed were appropriate or adequate.
"The board is of the unanimous view that a clear, organisational message needs to be conveyed that inappropriate behaviours can simply not be tolerated.
"The chief constable has now been asked to ensure the investigation that is already under way is completed expeditiously and a further report is provided to the board when complete."
Mr Hamilton said the ongoing investigation, led by Ass Ch Con Alan Todd, would "report to the Policing Board in due course".
Everton have completed the signing of 17-year-old Bury defender Matthew Foulds for an undisclosed fee.
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The Northern Ireland Policing Board has said that sanctions imposed on student officers who allegedly cheated in exams were not "appropriate or adequate".