Going by that stats for nerds screen, YouTube is receiving plenty of throughput going by its measurement. If it's correct, 26Mbps is enough for even 4K streaming, not that I suggest trying that over 4G.
The long delay is likely a problem with the CDN it is trying to connect to. YouTube first tries connecting to a nearby CDN such as within the Three network. If that fails, YouTube will then stream from another CDN.
The first thing I suggest is try setting the DNS on that computer to Google's:
Right-click the Wi-Fi icon in your taskbar and then "Open Network & Internet Settings"
Click "Change adapter options"
In the list of networks, Right-click the WiFi icon and then "Properties"
Click "Internet Protocol Version 4 (TCP/IPv4)" then click "Properties"
Choose the option "Use the following DNS server addresses:".
Enter 8.8.8.8 for the preferred and 8.8.4.4 for the alternative.
Click OK to both screens.
Try playing a few YouTube videos to see if they start any quicker. If not, you can undo these steps by following them, but choose "Obtain DNS server address automatically" for step 5.
Another thing you can try is run the utility DNSQuerySniffer, which you can download here. Choose your Wi-Fi Interface in the list and let it run as Administrator (required to sniff traffic). Start playing a YouTube video and check the DNSQuerySniffer screen for any hosts ending in "googlevideo.com". The following shows an example from my end:
If any of the googlevideo hosts have a red dot, this indicates a failed look-up. You can try also pinging the IP addresses under the 'A' column to see if they reply, particularly for the last googlevideo.com one it shows when YouTube is still at the black screen.