How long does it take for that annoying little voice that says “give it a miss” or “do it later” to go away?

In my experience, it doesn't ever really go away. You just have to learn to not listen to it. Something that helps me is to tell yourself that if you complete this task, you'll give yourself a reward immediately after. For example the task could be exercising, reading, doing work, and the reward might be a snack or watching an episode of your favorite show. If it's a long task, divide it into 30min to 1hr sections and take 5-10 min breaks.

It takes time to actively stop this from happening. You have to question your reasons as to why, deep down, you don’t want to do something. Is it out of fear? A lot of people procrastinate out of fear of failure and many people miss social events due to social anxiety. This won’t go away unless you address the root of the problem. Therapy is a great way to help with this.