Hello and welcome to MedHelp's forums. We appreciate your question but are sorry you have this to worry about. This is a question you can ask your doctor. However, I do want to point out that leukoplakia does happen for other reasons than oral cancer. In fact, oral cancer is rarely involved. But it is still a possibility and your doctor is being thorough to rule it out. With your family history and that you had lip cancer, that increases your risk. I can understand your concern. Here is information about leukoplakia in general. https://www.webmd.com/oral-health/guide/dental-health-leukoplakia#1. Please note that usually leukoplakia is harmless.
I understand wanting to have this addressed sooner rather than later. Again, it has to be noted that leukoplakia while it may not be cancerous puts you at greater risk (even if that is still minimal) so good medical care is important. Is seeing a doctor immediately in a private practice something you can do financially?
As to how fast it would get worse change stages if it is cancer, that is individual for everyone. Hopefully you have been advised to avoid drinking alcohol and stop smoking if you do either of those. How quickly this type of cancer grows depends on different factors. https://www.hopkinsmedicine.org/health/conditions-and-diseases/oral-cancer.
For most people, I would think they could wait. However, you have a history that puts you at greater risk. How long until you can get into the public doctor?
I certainly understand the finances of this and know that can be difficult. You have to follow the route that makes the most sense for all the factors in your situation. If there are any changes, I would tell the office you are on the wait list for. They may move you up the list sooner. I'm glad you gave up some of the risk factors. Stay in touch with us. We wish you the best.