Just to clear something up. Routers are the device that route traffic between the Internet and your home network. When talking about home routers they tend to come with a modem built in that connects it to the Internet (dealing with signal from the box down the road or the exchange, which is often the cause of poor Internet speed and dropped connections, rather than WiFi signal), plus also usually come with WiFi access point built in but often aren’t very good and lack a decent antenna. Because of their design as the modem they are usually located by the phone socket which is often in the worst place in the house to create a good WiFi signal.
The WiFi part of it may be dodgy and replacing the router with something different may get you a better access point and WiFi signal, but it may introduce other issues in the routing and the modem part, plus if not from the provider will require configuration. Though might resolve issues with the Internet connection itself if there are any.
WiFi extender all depends on the kind of extender. Many are crap. Some work okay, some work well.
A good Wireless Access Point can be wired from the router to a central point or act as extender. Generally better than the little extender things that plug into a power socket. Have a look at TP-Link access points. Worth spending a bit on a decent wall or ceiling mounted one. They usually offer 5Ghz as well as 2.4Ghz. 5 is a less congested space and performs well over a short range, but it’s poor over distances. 2.4 will go further.
Mesh stuff is basically the same and access points can act in a mesh. Just allows you to use multiple ones.
Aside from that, Talk Talk are one of the worst providers in the country, so changing provider may solve a lot of problems, even if it’s just support.