The reason why TT have switched you to another company is that people like you without a fibre connection drag down their customer average speed statistic. They will eventually advertise that TT customers have the highest average speed in the hope it will attract more business.
I don't think BT can stop any ISP putting in their own equipment in BT exchanges but in smaller exchanges the ISP's will pick up few customers so the investment is not worth their while.
It is unlikely your speed would increase with another ISP anyway but another ISP could give you a lower bill.
The issue for you is the distance you are from your telephone exchange. The longer the distance the slower the speed. Openreach are putting in fibre cabinets next to the existing green cabinets seen on many street corners. This basically means that your internet speed is given a massive boost because your copper connection only has to travel a few hundred metres and not several miles.
The question you have to asked is how long, if ever, will your local cabinet take to get a fibre connection. There are BT websites where you can put in your phone number for an answer like this one
https://www.homeandwork.openreach.co...get-fibre.aspx