Let It Be was nearly completely recorded before Abbey Road, with some recording sessions made in early 1970 (when the band was pretty much already split, as there was never all four of them together again after the Abbey Road recordings).
I think I Me Mine was the only Let It Be song recorded after Abbey Road, and that song doesn't have Lennon on it (as he had left the group at that point).
There is an error in el tigre's post though - Spector came in after Abbey Road was made, not before. The Get Back sessions (what the Let It Be album was originally called) were somewhat of a disaster and the album made from them at the time was scrapped, and the band rebounded with their (in my opinion) 'true' final album, Abbey Road.
It was only after Abbey Road that Spector was brought in in an attempt to work on the Get Back stuff, and remixed pretty much all of it. The Let It Be... Naked album released about 10 years ago is a reworking by McCartney to get it to sound more like the original vision.