I'm often asked how long it takes to change speech. There's no answer for that. How long does it take to learn how to play the piano? It's a similar situation because it all depends on the individual person. Two of the most important variables are motivation and opportunity.
Motivation = Why do you want to learn this new skill?
First, Is it your choice or are you being forced to do it? I've had students in my accent modification classes who were required to take it and who didn't want to be there because they said it wouldn't work. They were right. I've met people who never took a class about pronunciation or accent because they thought they could do it all by themselves. They were right. Second, motivation can come from internal feelings of wanting to do something better and better or external situations that reward better performance. I've been inspired by the internal motivation of retired grandparents whom I've taught that had a strong desire to continue to learn, to improve, and challenge themselves although they had no immediate need to perform well in English. The external rewards are all around and vary from having the communication skills to meet people and make more friends to getting a dream job or promotion. If the motivation is there, it will find the opportunities. Lack of motivation finds excuses for why it can't be done.
Opportunity = Use it or lose it
Motivated people create opportunities to consistently practice and the effort they put into practice produces results. I could be describing changing speech, learning the piano, or losing weight, it's all the same. When you can find something you enjoy doing as your way of practice, it's easy to keep your motivation levels up. I'm often asked what is the best way to practice speaking English and I reply with a question about what do they enjoy doing. There are many ways to find other people who share the same interests, Meetup.com is a good one and volunteering is also a good option, try Volunteer Match.org to find organizations that you're interested in. If you don't want to be bound by location, there are online conversation sites as a way to practice English with other people such as Verbling.com and WeSpeke.com. Follow your interests and you will have something in common with the people you meet and you'll have great conversations not boring practice exercises.
There are two sides of learning: concept and application. I can teach you the concepts and you can prove to me that you understand them, but can you apply them to your speech to try speaking in a different way? This is where most people get stuck in making lasting changes to speech because they let fear creep in. Fear of how they will be perceived becomes stronger than the motivation or opportunity. My advice is to have fun experimenting, don't be afraid to make mistakes, and each attempt will lead you closer to better performance because you're practicing. If you learn the musical scales but never actually play a song on the piano, how can you expect to improve? Find your song. Find your voice.