Quote 1:
"A PPL licence is required when recorded music, including radio and TV, is played in public. There is no statutory definition of 'playing in public' (also sometimes referred to as 'public performance') but the UK courts have given guidance on its meaning and ruled that it is any playing of music outside of a domestic setting – so, for example, playing recorded music at a workplace, public event or in the course of any business activities is considered to be 'playing in public'. In contrast, any recorded music being played as part of domestic home life or when there is an audience entirely comprised of friends and/or family (such as at a private family party) does not require a PPL licence."
Quote 2:
"(Q:) Do I need a licence from both PPL and PRS for Music?
(A:) PPL and PRS for Music are two separate independent companies and in most instances a licence is required from both organisations for you to legally play recorded music in public. While both organisations licence the use of music and collect royalties for the music industry, each represents different rights holders and have separate licences, terms and conditions"
Source for both quotes above:
http://www.ppluk.com/I-Play-Music/Businesses/Why-do-I-need-a-licence/#doineedalicence
Quote 3:
"If music is ever played on your premises for customers or staff; for example, through radio, TV, CD, MP3 or computer speakers, or at live events, this is considered a ‘public performance’ . The Copyright, Designs and Patents Act 1988 states you need to get permission from the copyright holder to ‘perform’ music in public – and a music licence grants you this permission."
Source:
http://www.prsformusic.com/users/businessesandliveevents/Pages/DoIneedalicence.aspx
So you probably require two licences in addition to your TV licence (which only gives you permission to install or use a TV set, not for a 'public' [i.e. 'non-domestic'] performance of music. However if you were only, for example, to watch the BBC News Channel then (or other non-music channels) no additional licences would be required.