Main Building (Cardiff University) as St. Luke's University
The Doctor calls Bill in to his office and arranges to tutor her.
Source: Doctor Who — Tumblr
“The Pilot” Filming Locations
University employee Bill is being tutored by the mysterious Doctor when a mysterious puddle leads to the disappearance of the girl she likes.
Doctor Who episode “The Pilot” was filmed in Cardiff in the United Kingdom.
Cardiff City Hall as St. Luke's University (clocktower)
Bill looks up at the clock and sees it is time to visit the Doctor.
National Museum of Wales as St. Luke's University (lecture hall)
The Doctor gives a lecture to a classroom full of students including Bill and Heather.
Cardiff School of Management (Cardiff Metropolitan University) as St. Luke's University (cafeteria)
Bill works in the cafeteria and smiles at the girl she is serving.
Source: The Doctor Who Locations Guide
Clwb Ifor Bach as Bar
Bill is out at a bar when she spots Heather sitting nearby and they approach one another.
Source: The Doctor Who Locations Guide
Sir Martin Evans Building (Cardiff University) as The Puddle
Heather leads Bill to a puddle between a few buildings that contains a very odd reflection.
Lookout Cafe and Bar as Sydney Restaurant
The Doctor takes Bill to a cafe in Sydney to escape the pursuing water only for it to catch up with them in the bathroom mirror.
Taff's Well Quarry as Far End of the Universe
Bill, the Doctor, and Nardole head to the far end of the universe to escape pursuit only to find the puddle there waiting for them.
Source: The Doctor Who Locations Guide