Some can afford to pay privately..... I've represented independently wealthy asylum seekers. But they are the minority, because even people who were moderately comfortable in an asylum-producing country won't have savings which go very far here.
The person has to find a legal aid solicitor who has space for them (not easy if they are in a "legal aid desert" and so some try to represent themselves). The solicitor then accepts them for legal aid and deals with the application up to the decision stage.
If they are refused, then the solicitor has to decide whether or not to grant CLR ("controlled legal representation") which is legal aid for the appeal stage. To grant CLR the appeal must have above 50% prospects of success AND the client must meet the means test. If the solicitor thinks it's hopeless, they won't get CLR.
"What stops the solicitor just granting CLR to everyone claiming they all have a 75% chance of success?" I hear you cry. Well, the Legal Aid Agency come in and audit them, and if they haven't got a proper success rate then their contract gets taken away, so a fairly huge incentive to do it properly.
The solicitor then gets up to £1600 to prepare the appeal. As a minimum that is a full asylum appeal statement responding to each of the issues raised in the refusal letter (the refusal letter generally runs to approximately 50 paragraphs), and must produce the statement in the appellant's own language with a certified translation. They must organise a "bundle" of evidence including any evidence the client is able to produce (with certified translations) and they must go through all the country background material and put in anything needed. And the bundle needs to be properly indexed and paginated.
I get a £302 fixed fee for the hearing. That includes the preparation, usually something between 3 and 6 hours depending on the size of the bundle and complexity of the case, then going to the hearing, seeing the client and going through procedure, and representing at the hearing - so in total, about 1.5 - 2 days work. That is before tax, NI, travel and chambers' rent.
So yes, legal aid is paid for by the taxpayer - I give you all the detail because I think the taxpayer gets a relatively good deal. 🙂