The answer is yes and no.
The game treats both worlds as 'Home-Worlds' while your Sim is residing there. In other words, things which are done in Oasis Landing stay in Oasis Landing. This applies to self-employed careers as well. While in Oasis Landing, any goods your Sim sells will not be counted towards the career goals which they obtained in their original home world.
I learned this the hard way when I had a self-employed painter from Sunset Valley create and sell a painting in Oasis Landing. The result was two (x2) self-employed painting careers. The progress of each career was anchored to whichever world they happened to be in at the time. In 'The Valley' she was level 9, in OL she was level 3.
While I haven't had a Sim join any 'official' careers while in Oasis Landing, I don't see this changing anything. I guess the only way to really find out is to test it. Logically, it shouldn't affect anything because technically Oasis Landing doesn't officially exist (yet/while in the original home-world) so anything which happens there is time fluid.