OK I want to add few things to this post in case it helps others. First I would like to say that the suggestion on using a "managed" switch is correct although it may be an over kill. The distinction of managed and unmanaged switches has long been blurred. This is mainly due to cheap and powerful SoCs (CPUs) which now can be added to switches with little impact on the price. If you look for definitions you will see that an "unmanaged switch simply allows Ethernet devices to communicate with one another, such as a PC or network printer, and those are typically what we call plug and play. They are shipped with a fixed configuration and do not allow any changes to this configuration". A managed switch "typically supports configuration, SNMP, remote access, vLANs, port mirroring, and QoS". So I think it will best for anyone looking for a switch to solve this problem to check if the switch supports both IGMP Proxy and IGMP Snooping, rather than being managed or unmanaged. I have a switch which was sold as unmanaged but has a simple web interface that supports vLANs. And that brings me to another point, you could have solved the problem you reported with vLANs as well. Most decent routers support vLANs as well. In fact I had IGMP Snooping enabled on my router but this still meant that the multicast traffic was being spread across my whole network. The reason was due to my network topology. This is how my devices were connected:
BT Openreach VDSL modem => Asus Router => 24 port Switch => all other devices including BT TV Box
My Asus router supports both IGMP Proxy and IGMP Snooping. The way IGMP Snooping works is like this. The router will send multicast packets only to those ports where multicast traffic is requested by the multicast device (i.e. your BT TV Box). However as you can see from my network topology the BT TV Box was connected to a 24 port Switch and then that switch was connected to the Asus router. So even with IGMP Snooping enabled the Asus router was still sending multicast traffic to the switch (as the BT box was requesting it via the connection to the switch) which in turn meant the switch will broadcast it to all my network devices connected to it. BT has recently (Nov 2017) made a change where they now continually broadcasting in multicast some programming data which they say will be used to improve recordings when events overrun (i.e. sport events, etc). See this thread for more info. When I installed BT TV I did not noticed any issues with the multicast traffic being spread to my whole network, but probably just because I was only occasionally watching BT Sport. After they started to continually broadcasting in multicast programming data some of my wifi devices started to have wifi issues where they will drop out of the wifi network. I have a mesh wifi system (Ubiquiti AcPro) and the management interface showed that there was constant traffic on the wifi network. After some research I nailed it down to the multicast traffic of the BT TV box. But due to the change BT did even switching the BT TV box off it would mean that the continuos programming multicast traffic will go to all my network devices, even the wifi ones. While BT claims this constant multicast traffic is "low bandwidth" and won't "affect anything" I beg to differ given the problems I found. The solution for my problem was as follows:
1) Run a separate network cable between the Asus Router and the 24 port Switch. This would allow the Asus router to correctly do IGMP Snooping and send the multicast traffic ONLY on the second port/cable to the switch, not the original cable that connected the router and the switch.
2) Create a vLAN on my switch to separate the traffic for this second cable between the router and the switch. The vLAN was formed of two ports only: the second port/cable to the switch and the cable/port for the BT TV box. This meant that when the switch got multicast traffic via the second cable from the Asus router it wouldn't broadcast it to the rest of network devices connected to my switch as the vLAN was isolating it and it was only being sent to the BT TV box.
This might sound like a complicated thing to do but is relatively easy to set up if you roughly know what you are doing. So in summary if you connect your BT TV Box directly to your BT Home Hub (or any other router which supports IGMP Snooping you won't see the problems I am describing. But if you connect your BT TV Box via a switch/hub or your network topology is different this post applies to you. And this issue is even more critical if you don't use your BT Home Hub/wifi router for wifi and have separate wifi access points connected to a switch. So you need to see what's your network topology and what is the right solution for you. vLANs, IGMP Snooping, or both might be needed. If you use a switch you might need a switch which supports vLANs or IGMP Snooping or both. IGMP Snooping is more CPU intensive than vLANs, although most recent decent routers and switches should be able to to do IGMP Snooping without much effort.
The most clear indication that multicast packets are being sent tro your whole network is that you will see ALL of your switch port LEDs constantly blinking and never rest. They will also all blink at the same time which is indicating the same packet is being sent to all your switch's ports at the same time. Final tip is that if you make changes to your network topology which impact IGMP Snooping you may need to reboot any devices (routers/switches) doing IGMP Snooping for them to readjust to the new topology.
Hope this helps other people!