Hey mate,
Firstly, I know next to nothing about engineering so I'm just looking at this from a rational point of view.
The undergraduate degree is 4 years long because I'm assuming that they teach you everything from the beginning. For the first year (maybe two) you would study general engineering and you would then go on to major in something (ie. Civil engineering).
In regards to the Masers of Engineering, I'm not sure what the prerequisite subjects would be or what the application process for your chosen field would be like but there would definitely be a lot of assumed knowldge. Hence, the 2-3 year full-time course. Take a look at this, for example.
http://www.eng.unimelb.edu.au/study/graduate/master-eng-biomedical.html.
I doubt many people would go into a Masters of Engineering from a completely unrelated background because they may not have satisfied the entrance requirements. That's upto you to look into.
I don't think specialist maths is a prerequisite (even for undergraduate engineering) but it would probably help.
I hope that was helpful in some way!

Edit: mark_alec bet me to it.