I have been getting these calls for weeks and I have sometimes been rude and sometimes taken the trouble to explain that I have it on authority of the real Talk Talk that they are trying to scam me. Now, I just ask them for their full name and they immediately ring off. Up to now I have not had any problems with my internet and I suspect that your problems are nothing to do with these scams.
I would insist that T/T send out an engineer to check your line. You might have to pay if they don't find a problem but, as you obviously do have one, that shouldn't be a problem. The engineer will be from Openreach so not biased in favour of Talk Talk. In fact T/T will have to pay for the visit if there is a problem so they will ensure that they do all they can to check your line first. I find that the chatline is the best way of getting T/T to do anything. At least then you don't have to worry about understanding the Indian accent.