I will just be a different voice here, just to hopefully put your mind at rest a little.
The very worst case scenario you will face is a criminal record and a fine, linked partly to your income. In addition to the fine, as a punishment, there is a victims' surcharge, court costs and TfL's costs. TfL will also seek to recover the fares that you did not pay. That may come to a lot of money, but the courts will allow you to pay it in instalments if you cannot afford to pay it all at once.
Though technically there is a possibility of going to jail, it just doesn't happen. Not right now when the prisons are full anyway, and not anytime. You will not go to prison at all. You just need to cooperate with the system.
That was the bad news. The better news is that this is a problem, but it's not the end of the world. It's a low level crime, you will have to accept the punishment and after a while you will be able to move on. So long as you don't act dishonestly again, all will be OK eventually.
In terms of visa advice, you will have read the forum's position on that. There is guidance on the issue on the website of a respected UK charity NACRO.