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One of the players in a mature group I'm DMing has a drow who is now pregnant. What kind checks should I have the player make and what kind of changes and modifiers for her stats will I need and when(chronological guesstimation)?

Or, you know, just handwave away the gestational period.
"It's been [time].  The drow has a kid now."

One thing to consider is that a Drow pregancy is unlikely to mirror a human one. Drow, however different, are a type of elf and as such share the fae characteristics in terms of their bodily functions. A race that does not require sleep and lives for centuries will likely have a very different gestational process than other races. I don't know if there are any flavour texts that lay this out but the pregnancy may last for months or years, the mother may be significantly hampered or may feel very little effect. Perhaps the gestational period lasts for years and only in the very last stages does the mother begin to "show" and feel physical effects. As the DM the details are up to you and how you want this to effect the party.

Do you want to get it out of the way and have the baby left with a wet-nurse or do you and the player want the drow to go through an extended pregnancy and child rearing process? In the latter case maybe they should be retired to NPC status. How you handle it really depends on how the player would like their character to handle it along with how you feel an elven pregnancy should physically manifest itself in the game. If it's going to be a pregnancy physically similar to a human one then there's just no way the character would be able to keep adventuring. Several close friends have recently gone through pregnancy and none of them seemed likely to be swinging a sword (spell, spear, bow, whatever) around with any kind of alacricity.

If the player is determined that the character would try to soldier on then you would expect gradually increasing penalties to dex and strength, armour to stop fitting and maybe even movement speed to be reduced. In the last stages maybe even applying a permanently fatigued or exhausted state to the character to represent the mobility and other issues that heavily pregnant women encounter. Definitely steer clear of trying to impose any mental (Cha, Wis, Will) penalties based on this. Different people handle pregancy very differently in terms of its effect on mood, etc. If you player is as mature as you say, they will RP this better without rules based inflictions. Again though, all of this is dependent on how you as a DM want to rule that a drow/elven pregnancy actually works in comparison to a human one.