This is what ping is for. Though a tool like smokeping (no idea if that can run on Windows) that plots the results of ping tests over time is probably more useful for diagnosing intermittent faults.
Obviously that will only show you what is or isn't reachable, not why, but you can infer a lot[1] if you see packet loss between your laptop and your own router, for example. Just leave a ping -t running in the background and flip to that terminal window when you notice a problem. Start with something on the wider internet, like google's 8.8.8.8 DNS server. Then try something within TalkTalk's network. Then your own router.
I've no idea whether or how Windows can provide system logs that show what its network connections are doing. That could be useful for working out the nature of a WiFi problem.
Your router's system log may provide access to the state of the PPP connection, which can be more revealing if it's a broadband issue. (Has LCP timed out? Has the IP address changed? That sort of thing.) Talktalk are presumably too cheap to provide any useful diagnostics from their end. Similarly, there might be WiFi access point logs (associations, authentication, roaming, that sort of thing).
[1] Eg. If it's a WiFi problem, then it's nothing to do with TalkTalk.