You will need to take exams, both in just maths and in other subjects which require a great deal of mathematical work, in any engineering degree. It is a necessarily mathematical field, and you will need to use maths at A-level and beyond not only in your mathematical methods modules on an engineering degree, but in almost every other module - you will be using calculus, solving differential equations, doing matrix algebra etc, on a daily basis, across all your modules in an engineering degree. If you cannot do A-level Maths exams you will fail engineering degree exams, in just about every module except the non-mathematical (and easy) engineering management/professional skills modules. There really is no way around that. You need to plan to study another subject if that is the case for you, or resolve the issues causing you to fail in maths exams now.
For medicine you can find details of all the entry criteria for all UK medical schools, and innumerable queries and discussions about subject combinations, in the thread created by
@ecolier here:
https://www.thestudentroom.co.uk/showthread.php?t=5611422 you will generally need A-level Biology
or Chemistry plus usually another science or maths subject (which may be whichever of chemistry or biology you haven't already taken, physics, or maths, plus sometimes psychology is considered). Some require both chemistry and biology. One (Newcastle) has no subject requirements. None require A-level Maths.