How fast you can learn enough programming to get a job?

I got a job, at a major mining company, with no qualifications and very little experience because I was able to walk into an interview, slap down a thick listing and say I can debug and fix code like this. The listing was some horrible BASIC code from an accounting system I'd been working on part-time after learning a bit of how to program at the University Computer Club (it was in the early 80's).

Fixing other people's problems is a great way to learn and to get credibility. I think someone working hard, with reasonable intelligence and a good eye for detail, can learn to do this in a couple of weeks. You need to do enough to build up credibility and to be sure you're coping with a range of things. Some of the research below alone will also take you a couple of weeks.

My suggestion is:

1. Pick your field (probably web development)
2. Find programmers in your field either nearby or associated with companies of the kind you want to work for.
3. Find out what open source code they use.
4. Review the bug lists and languages for that source. This alone will start giving you things to study, as you try to understand the bugs and you may find you quickly learn which language feels more natural to you.
5. Pick something to work on based on it having a lot of bugs for you to fix, so you can be of value to that community, which are of varying size. You should plan to be a member of this programming community for years - it's an important part of your career credibility. So, pay a bit of attention to the tone of the email conversations.
6. Fix a bug, submit it, repeat, build up reputation.

You can find people and what they use by starting with searches for company names, look on Linked-in to identify individuals and then search for their postings on programming forums. Use Find your people - Meetup to find discusssion groups that meet in person in relevant technologies and go to a few meetings across different related tech, both to listen and to meet people. 

Tell people what you are trying to do and also ask if you can get contact details and you get back to them with progress - most people are interested in seeing how people get on and this is a way to build a reputation as someone who delivers.