An interview in the UK Sunday Times revealed that actor David Tennant was originally hired to direct this TV special, but had to drop out (being replaced by David Kerr) due to personal life issues related to Tennant's wife's pregnancy. Tennant still acted unofficially as writer Russell T Davies' personal 'Shakespeare expert' during the time he was working on it.
The street seen near the beginning was used in Dr Who (Face the Raven & The Husbands of River Song)
The fairytale forest was filmed in Puzzlewood (Dr Who, Star Wars: The Force Awakens).
When Lysander sings the song "Tomorrow is Saint Valentine's day" to Hermia, the CCTV footage clearly shows that it is the night of May 29th to May 30th. Therefore, the Saint Valentine reference is quite off here.