asylum seeker is person who is seeking asylum and not who was already granted asylum. Azylant in Czech means a person who was granted asylum. Žadatel o azyl (or coll. uprchlík) is person is seeking asylum = asylum seeker.
I don't entirely agree with this comment. An azylant has been granted time-limited asylum status for the period specified in the interim decision on their case. Their claim for permanent asylum has not yet been finally assessed.
azylant - cizinec, kterému byl udělen azyl, a to po dobu platnosti rozhodnutí o udělení azylu. Source: Czech Interior Ministry
Uprchlík is any kind of refugee and the word is not colloquial. A refugee is not necessarily an asylum-seeker.
Different countries have different ways of describing asylum status. In my view, "asylum-seeker" would usually be a reasonable translation for azylant in many contexts, such as this police report, where an asylum seeker (described as an alien who has sought international protection and is in the Czech Republic legally) with a long record of crime will be deported "only if a refusal of his asylum application takes legal effect". So his current asylum status is "interim" or "temporary", not "definite".