I don't have the links at hand right now, but my wife looked into this extensively both for cribs for our first and for painting our then-newly purchased house.
- Purchase a solid wood (not particle board or any other glued-type processed wood) crib. Varnish/paint doesn't seem to be very flexible; any crib will be finished, and the natural wood cribs will have VOCs in the finish just as the white/etc. painted cribs will - often more. (Wood finish/varnish is worse than paint in most cases.)
- Purchase one made in the USA (such as Land of Nod's Andersen cribs, which is what we went through). USA (or Western European/EU) maker will likely use relatively less dangerous paints than in some other countries.
- Well ventilate the room so VOCs do not build up in large quantities. Don't forget, your paint on the walls also contains VOCs likely, as will your floor if you have hardwood (and carpet has its own fun things!).
- Buy a mattress protector that fully encloses the mattress.
Most of the noticeable by smell VOCs leave fairly early on in the life (within a few weeks after painting/varnishing - and much of the noticeable by smell isn't actually VOC, so don't assume that is the case), if it's done properly; if you're still smelling them likely months+ after it was painted, it may be a sign of a bad painting job or some other problem.
You could consider buying a used crib; the VOC level drops over time, after around 5 years you won't have very many VOCs given off comparitively, and can easily re-paint with a safer paint. I assume you're not in the US from your word choices, but if you are, don't buy a crib made before 2011 (which admittedly gives you a fairly narrow range of dates...) as they aren't legal to sell due to safety concerns (see this link at the CPSC for more details).
If you do the travel cot option, I don't necessarily think a softer mattress is a good idea. Harder is better for SIDS purposes, and most kids that I've known are even very comfortable sleeping on hard wood floors. I'd also not assume it gives off nothing dangerous; it's probably largely plastic wrapped around metal legs, and both the plastic and (less often) the metal can be significant sources of risk. BPA wasn't thought to be dangerous for a while, nor were pthalates, so who knows what next be found dangerous in plastics. Properly finished/treated wood should be okay.
You can also remove the finish from the crib and re-apply a low-VOC paint of your own; this isn't all that hard to do (something like a 6-8 hour job to remove, then a few hours of work plus waiting to dry for re-finishing/painting). Paint is better than varnish if you have the choice.