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.