Better to start your own thread to avoid replies getting confused.
Pixellated pictures on live TV are often the sign of a signal fault. If you've rebooted your box once, then there's likely a fault somewhere, so this will need VM to intervene. You're likely to need either an engineer visit or a replacement box, depending on what VM's remote diagnostics turn up.
If BBC2 HD is affected, check BBC1HD as it's carried on the same frequency and should be similarly affected.
First check the automated faults line for any existing issues in your area that could be the cause: 0800 5610061. If nothing's reported, assume VM don't know about your fault and ideally you'll need to call them.
150 from a VM phone, 0345 4541111 from any other phone - choose the options for "I have a fault with my TV service". Alternatively you can wait on here for staff to respond, which may take a day or so.
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Thanks for taking the time to raise your TV issue and the problems with BBC 2 HD.
I've taken a look at your connection and equipment and everything appears to be within the spec's and ranges we'd expect to see. One point we would like to raise, that may not be associated with the TV service issues, the Hub is currently showing as being online for 81 days. We'd recommend regular reboots to help refresh the connection.
In relation to the TV issue, can you follow the guidance from japitts and check other channels in HD like BBC 1. Also, if you have more than one set top box, do you experience the same issue across all boxes?