I would like to publish an article open access, but I cannot afford the article processing charge (APC). I am based in the UK, so I do not qualify for some of the available fee reductions or waivers. I also do not have grant funding to coverage the fee. How can I obtain funding to cover the publication fee to make my research open access?
Where are you based? If you're at an institution, I'd get in contact with your library. They usually have funds of some sort to cover APCs these days.
Depending on where you're submitting, it is still possible to get a fee waiver if you can provide financial justification (which it seems like you should be able to do). I don't know how well this works at most journals, but those like PLOS ONE and PeerJ are well known for providing fee waivers for researchers with OA support (apart from a couple of incidents).
Alternatively, you could pursue the 'green' route of OA. Submit your paper, but don't make it OA (via the 'gold' route), and self archive the pre-print, post-print, or whatever the options available for your journal are. You can check policies here: http://www.sherpa.ac.uk/romeo/
Try playing with these options a bit, and then report back if there are further complications or details to add.
When preparing your manuscript, you could also approach the journal editors and ask for their policy on waivers. I would do it before finalising the manuscript according to the journal's style. Even if the journal does not offer waivers, journal editors should be interested in finding out how often APCs are preventing people from submitting to their journal.
You could co-author the article. Authorship will be shared so does cost. Perhaps co-author could have such cost free options. That will not only minimize your cost but will open you to collaboration. Of course there is a trade off in such a scenario but better than delaying the publishing or even not being able to publish.
Please look at my comment below. I did not mean it something unethical.