Moderators - should this thread perhaps be moved to the UK/Ireland forum, seeing it's specifically about the UK?

For the OP - as already mentioned, it's your employer who will have to apply for your work permit. The general rule is that a work permit will only be granted if the employer can show that they tried, but failed, to fill their position with someone who already has the right to work in the UK - i.e., a UK citizen, a UK permanent resident or a citizen of a member state of the EU.

If you have already been offered a job then this probably means that your employer is fairly confident of being able to get you a work permit.

A friend of mine had UK work permits on three separate occasions, and from what I can remember it took a few weeks for the paperwork to be sorted (i.e., it wasn't a case of a few days, but neither was it a case of having to wait for months on end).

Good luck!