A handcuff can't be picked with a finishing nail (like the one Wyatt pulled from the chair) without it being bent towards the tip. When Lucy retrieves it from the trash can, the nail is straight. When Wyatt uses it (for the second time), it is still straight.
Contrary to what is implied, Nixon did not record conversations secretly. The recording of meetings was a common practice in the White House, and something that Presidents Kennedy and Johnson also did.
When Rufus is given the number to call in 1972, it reads "JK5-0168", which translates to the fictitious phone number "555-0168". However, JK5 is not a proper exchange name, which were used prior to the 1960s. The correct telephone exchange name for 555 is KL5 or "KLondike-5".
Just after Lucy & Wyatt escape Flynn's hideout, there's an aerial shot of D.C. from the Capitol to the Washington Monument. The Smithsonian's American Indian museum, the Hirshorn museum, and the Air & Space museum are prominently visible, even though they didn't exist at the time this episode takes place.
Lucy tells Rufus that besides Woodward and Bernstein, nobody knew Deep Throat's real identity until years later. In actually, screenwriter Nora Ephron, who was married to Carl Bernstein, figured out who he was - and after their divorce, when asked, would tell people it was Mark Felt. Nobody but her two sons believed her, though.
At the end of the episode, a Polaroid SX-70 photograph was shown. This camera was not in widespread use until 1973.