Its just the poor performance of BT Home hubs wireless, I use separate wireless access points with my home hub 1, as that one (the very first home hub) suffers from the same issues. Its not a signal level issue as the HH1 has an external aerial, its just that if you try to stream continuous data over wireless, it simply gives up and disconnects.
I don`t think any of the later home hubs are any better in that respect.
Is it practical to run a temporary Ethernet cable from the home hub to the TV?
If not, then Broadband extenders are going to be the next best solution.
These are specifically designed for that purpose. You can get them from Amazoon as well as the BT Shop.
That would be a better option than changing the router at this stage.
I used to use a couple of TP Link ones, but they had a habit of going into standby and not waking up again. That was a few years ago, so there may have been a firmware upgrade.
I now use some old Comtrend ones, and they are fine.
You could always try a couple, and see if they work, at least they are easy to return if you have problems.
I see Argos do the BT ones. 498/9512