The WHO are cutting hundreds of positions. Unesco will have to do similar (due to US and Canadian policies on the admission of Palestine to the organisation). But the reason I say funding will be cut is that due to demographics and excessive debt, the rich countries of the world - the ones that fund the UN and other international organisations - are going to have less money to spend. And cutting money sent to the UN will be easy for them to do, politically. And the employees at the UN with reasonably secure contracts will not be moving on.
When I managed to break into the UN, it was due to an excellent reference from someone high up in the organisation. Without it, I'd have had zero chance. And 180 people applied for the position, even though it was for a 1-year contract in IT with a moderate salary in a not particularly popular city for UN jobs. And I only got the job because the person they really wanted (who was already working there) was not liked by the top boss. After a big fight, he finally forced them to take 2nd choice, me!

Most vacancies you see in the UN are not really open and are just advertised to meet some internal rules.
Other positions in this organisation had 600+ applicants.
Anyway, I don't want to put a damper on things, just some thoughts to explain why I say it's a hard road. You need some luck and a plan B.

A lot of UN jobs - even for volunteers - require 3-5 years experience, so you could also look to jump in from there. Also look up the JPO programme for your country of nationality.