I believe that part 3 is an optional 4th year for cambridge math students, who get their undergrad diploma after year 3 and can do an additional year if they so choose, but they also take graduates from other universities
from the Cambridge website, https://www.graduate.study.cam.ac.uk/finance/fees
(the master of advanced studies in either the department of applied mathematics and theoretical physics or the department of pure mathematics and mathematical statistics)
fees for home and EU students are GBP 10,065 and for overseas students are GBP 19,449.
there's something on that page about EU students and Brexit. You haven't said where you're from so we don't know if it applies or not, but at some as yet unspecified future date, EU students will have to pay the overseas rate
There are also college fees (on top of the university fees) which I didn't look for
Master of Advanced Study courses, which are equivalent of a fourth undergraduate year, are treated as undergraduate courses for fee purposes, and therefore overseas students are also charged the undergraduate College fee not included in the fee table below.
If you share more of your background, people might be able to help with suggestions for scholarships.
You haven't told us which scholarships you have already applied for, or where you're from, but if you're american and haven't already tried, try churchill scholarships
Will a bank (or other entity) lend you the money?
I don't know the rules, but it may be hard for an international student, since there's a risk to lenders that you go home when done and stiff them
Your best bet may be to get a job and earn enough to pay the tuition yourself.
I'm not as pessimistic as trackstar on deferral.
There may be a difference between an optional 4th year of an undergrad course (like part 3) and a stand-alone graduate course. I don't know the admission process for part III, but if it's at the college level rather than the university, you can probably just talk to the college and they'll let you do it a year later.