In the series 4 episode "Forest of the Dead", Donna is in the fake world and experiences time jumps. For example, quickly going from feeding ducks, going fishing, getting married and having children, and she starts to realise this. But shouldn’t everyone in the simulation realise this, not just Donna, so why is Donna’s husband surprised to find out that the whole world isn’t real?
The simulation is designed to make those inside it believe it, and 'Doctor Moon' seems to be an artefact of the program designed to smooth over any disbelief. Donna appears to accept it, at least for a while. If the "jumps" disorientate her, she manages to dismiss them. When Lee, reads Donna the letter saying "the world is wrong", Donna calls the sender a "nutter" (British slang for insane, crazy person).
Donna's realisation that she is inside an artificial reality is rather like 'lucid dreaming' - that is, once she is aware of her dreamlike state she is able to influence it and do something about it. Evidently, Lee did not come to that realisation, and everything the simulation is designed to do continued to work on him, as it did everybody else inside it.
Donna may only have been aware of the jumps in time because of her semi-lucid state. Alternatively, the jumps in continuity may only have occurred while the simulation (under the guidance of the doctor moon) was adjusting to suit Donna's dreams and desires. Once it had found the perfect simulated utopia for each individual, it may have settled down.
Really there is no reason that Lee, or anyone else from the library, should have become aware of their condition. Donna was exceptional in this regard.