No, they will not be able to work. And no, they will not be able to make an in-country application for a work permit (Tier 2).
Yesterday, 13 July 2015, the UK government issued a ministerial statement announcing a change in the rules affecting students (T4) who are pursuing a course at a publically funded college. The statement reads in part...
New students at publicly funded colleges will be prevented from being
able to work in the UK, in order to bring their working rights in line
with those of international students at private colleges. In the
autumn, college students will be unable to switch to a work visa or
extend their study visa whilst they are in the UK, whilst protecting
students at embedded colleges who will progress onto study at a higher
These new provisions will activate by the time Parliament rises in December. Transitional provisions for students whose applications are caught in the middle of the change will be published in about September or October.
As noted in comments, the UKVI site is now out-of-sync with the rules and it will take a while to get caught up. Students who are affected will need to read the legal briefing in order to stay current.
The full ministerial statement is at House of Commons: Written Statement (HCWS95)
See also: Annual Report