If I am still logged in using browser X, can a CSRF attack work when I use another (default) browser Y on the same machine? Assume that the web server has a CSRF vulnerability and no hidden token checking will be done on the server.
No. CSRF works by using a session already in place, and sessions can only exist within the same browser. Or put another way, all current/major browsers do not share sessions.
EDIT: As noted in the comment, it is theoretically possible using flash cookies. IMO, flash cookies are evil to begin with though.
Assuming that the existing CSRF vulnerability is based on the presence of a login cookie, then the simple answer is no (in the general case).
Different browsers (e.g. IE/FF/GC/Safari) each manage their own cookies, and logging in on one will have no effect whatsoever on the cookies for another browser.
However, there are other CSRF flaws which are no based on cookies, which may be relevant to other browsers.
For example, Integrated Windows Authentication -based CSRF would be relevant for IE, even if you had logged in on say FF. Or any other non-cookie based authentication/session management, including flash-based sessions, etc.