Your broadband speed is easily good enough for unbuffered catch-up - even in HD.
Unless your TV is quite elderly and slow, it's probably your wifi connection between the TV and the router that is the issue.
Is the router near enough to connect with an ethernet cable - even if it's not a long term solution, it will at least tell you if the wifi is the problem?
If so then you need to improve the wifi signal - try changing the wifi channel (if your router lets you do that) or try either a wifi booster, or using powerline adaptors.