My main character, Dumas Rilfore is half-English, half-French, from 1422. His background in Gothic Earth has him fluent in both languages. I've had him on the server for about 2 years, though I did take a nearly 1 year break with him until I started playing him consistently last winter.
Anyways, when I started him on the server, for ease of use, I just never mentioned the whole Common language barrier problem in my starting RP with him. Every player I ran into in those early days seemed okay with it, and I think everyone understood that a player may not want to go through the months that it would take to have them learn common. That's not fun for some people, learning a language like that.
After a couple of weeks, some real life stuff got in the way of any extended playing for me, and I took a long break. When I came back, the character base had entirely changed, so I used this as a chance to redefine my concepts for Dumas, and started RPing that he wandered the Mists for about a year, and picked up Common. Once again, everyone seemed cool with this. If you want to RP that your character did something similar, in my opinion, I think that would be acceptable. When it all comes down to it, this is a game, and this little bit of an RP crutch could be really useful for some players. It's not like Common is Draconian, or Balok, or Dementlieuse... it's just an ease of use thing, if you look at it from an OOC standpoint. I don't believe it is immersion breaking if you go this route.
At the same time, RPing learning Common for a Gothic Earth character could be fun. I can see that. It would certainly lead to some interesting RP situations, I'm sure! But it may not be everyone's playing style to do so. That said, if you want to avoid that long process, at least be sure to have a good RP backstory of how it happened, (relying on natural factors, not magic, I would suggest), if anyone questions it!
In the same vein of things, as my character Dumas is half-English, half-French, he knows the 15th century versions of both languages. There are times when I have him communicate as such (usually for a curse, or when he is surprised, scared, praying, or telling a secret to another Gothic Earth character). When that case happens, I do simply type, "[in his native English (or French)] Meet at the noon hour". Thus far, no character that does have an RP background of speaking English or French has acted as if they can understand him. At the same time, if Dumas runs across a Dementlieuse character, I don't act as if Dumas knows what they are saying. I'll perhaps have him feel more comfortable having that language spoken to him, due to his background. It has only been after spending considerable time in Dementlieu that I have had Dumas pick up a few phrases from that language. I believe it would make more sense for him to start to learn Dementlieuse than say, Balok. Now this isn't to say that it's because Dementlieuse and French share words (even if OOCly, they do, to give it familar flavor). It's more because it is supposed to have a similar feeling to the ear, sounding in tone and accent like French, not actual words.