I work at a UK University.
Here are the 5 texts recommended for first year Mech Eng students
Engineering Design Methods: Strategies for Product Design Cross, Nigel
Engineering Mathematics Through Applications Singh, Kuldeep
Introduction to Mechanical Engineering, An: Part 1 Clifford, Michael;Simmons, Kathy;Shipway, Philip
Principles of Modern Manufacturing Groover, Mikell P.
Thermodynamic and Transport Properties of Fluids: S. I. Units Rogers, G. F. C.;Mayhew, Y. R.
If you are more interested in design, we also teach Product Design. Here are the 4 recommended texts for that course
Manufacturing Processes for Design Professionals Thompson, Rob
Research Methods for Product Design Milton, Alex;Rodgers, Paul
Drawing ideas: A hand-drawn approach for better design Baskinger, Mark; Bardel, William
Prototyping and Modelmaking for Product Design Hallgrimsson, Bjarki