One of my players does not speak common as they come from a village that speaks entirely elvish. We are coming up on a couple weeks of downtime in the campaign and another one of the players wants to teach them common. According to the Xanathar's rules it takes 10 weeks to become proficient in a language, but they would learn some over a couple weeks. My question is how much of a language would a character pick up in that time span? How would they role play speaking common with a limited knowledge?
There isn't any rules for that.
Sometimes in D&D the DM is required to make a ruling, this is one of those times.
I have had someone try to learn a language using the 5e downtime rules, but I have played with players who have low int scores or are not proficient in a language. Usually, this involves the player giving some clunky broken English dialogue, and the DM further misinterpretng what they say. This allows finer control by the DM, since even if the player doesn't commit to the roleplaying and speaks in basic but acceptable English, the DM can still give them funny looks and have the NPCs misunderstand.
As for how much language is learnt in a given time span, I think making a rough guess would be fine. At 2 weeks they can speak a small, basic, amount and are frequently misunderstood. At 8 weeks they can speak very well but are occasionally misunderstood.
RAW, you either know a language or you don't
There are no rules to describe knowing a language partially, or being half way through training. As far as the rules are concerned, you either know it, or you don't.
If you wanted to represent a character learning a language over time, or having only learned some of a language, this would require a DM's ruling, without any RAW guidance. It seems reasonable that you would have some partial knowledge, since the RAW approach of "you either know it or your don't" isn't exactly very realistic, but that only means there are no rules to describe how that "should" work.
In Xanathar's Guide to Everything (p. 134), the rules for learning languages during downtime state:
Resources. Receiving training in a language or tool typically takes at least 10 workweeks, but this time is reduced by a number of workweeks equal to the character's Intelligence modifier [...]
Given that the Intelligence modifier is mentioned as a way of speeding up the learning process, the intelligence of the character would likely also impact how much of a language they'd have picked up in two weeks compared to someone less intelligent.
But roughly speaking, it's still going to be the DM's and/or player's interpretation of how much of a language can be learned within that percentage of the overall time it would take them (meaning, given that someone with an Intelligence score of 14 (modifier +2) would need 8 weeks, then two weeks worth of training would be 25% of the overall time needed, but what exactly "25% percent of knowing a language" means is up to the DM and the player).
I agree with the other answers that there is no specific rule for that, but if you want to house-rule it, you can offer the character Intelligence checks to see if they are able to communicate, with the DC set to whatever you feel represents the character's overall proficiency level. For example,
The guard is speaking very quickly, and your language training in Common is incomplete. Roll a DC 15 Intelligence check to see if you understand what he said.
The guard told his colleague, "I'm on duty until lights out, after which I'm going to bed. Be careful of that elf in Cell 3, don't let him cast any fire spells."
The guard said something about lights and being careful. You were not able to make out anything else.
On critical failure:
All you hear is unintelligible syllables. You can't make out anything.
You can have a similar rule for expressive communication. On success, you get your message across to the guard. On failure, he can't understand you. On critical failure, you inadvertently insult his mother.