The 20-year-old has joined the English Premier League side for a fee believed to be in the region of £12m, which could rise to £16 with add-ons.
The move was confirmed by both clubs on Tuesday afternoon, with the Scotland right-back agreeing a five-and-a-half-year deal until the end of June 2027 at Goodison Park.
Commenting on the transfer, Rangers sporting director, Ross Wilson said: “Personally, Nathan is someone that I’ve placed a lot of faith in since I joined Rangers. We saw top potential in him and I’ve enjoyed working with him and seeing him develop every day.
“Nathan is supported brilliantly by his family and as Rangers fans, I know they are extremely proud of his Rangers achievements.
“In terms of the detail of the transfer, I have been in discussions with Everton chairman Bill Kenwright for a number of weeks. It has been a pleasure to deal with Bill - a man who represents his club with genuine class.
"It is true that there were previous discussions last summer but we were always clear that in our model players will only leave at the right time and for the right number.
“We have now established a significant club record and there will be even further financial incentives for Rangers as Nathan’s Everton career evolves.
“We were very clear in these discussions that Nathan is a player and person who we hold in the highest regard and the deal had to represent a club record for Rangers.
“Since I joined the club, I’ve been clear that we would be strong in the market and our chairman and board have been prepared to reject some significant sums for our players, whilst we have all acknowledged that we would sell the right player at the right time for the right value. I must say in this case also, this is the correct destination.
“Our Academy staff can be proud of the role they’ve played in Nathan’s development. We are proud of Nathan, he leaves as a champion and I will follow every step of his career and journey and he will always be connected to Rangers. I am sure we will see him back supporting the lads at Ibrox soon.”
Rangers managing director, Stewart Robertson, added: “This is a deal which fits with our long-term, strategic vision for the club. We have been very clear that player trading is an important element of our business model but only when the circumstances suit Rangers.
“We are delighted for Nathan and his family, wish them all the best for the future and they will always be welcome back at Ibrox.”