Bartley was injured in an aerial collision with Forest substitute Dele Adebola and received extensive treatment before being stretchered from the field.
The Arsenal youngster was taken to hospital shortly after the final whistle and Speed revealed the extent of the damage in his post-match press conference.
"It's a fractured cheekbone," said the Blades boss. "We knew that straight away.
"You could see it was depressed - I saw from where I was. I've done it myself.
"Hopefully he will be back as soon as possible. Usually if there are no complications it's six to eight weeks, but it depends how bad it is."
Speed cleared veteran striker Adebola of any blame, adding: "It was a clash of heads. I know Dele and he's not that type.
"It was a bit late, Kyle got there first, but it was certainly an accident.
"Dele is a big unit, so it's unfortunate."