Actually battery powered TVs are not exempt, they are just covered by your home TV licence. If you don't have a home TV licence, then (IMHO and IANAL) you will have to pay. Also (as I read it) if you watch TV on your phone at some other location *and* have it plugged into the charger, it's no longer "battery powered" and so would have to have a licence at that location (except there are exceptions for caravans and various other things)
"It makes no difference how you watch TV - whether it's on your laptop, PC or mobile phone or through a digital box, DVD recorder or TV set - if you use any device to receive television programmes as they're being shown on TV, the law requires you to be covered by a TV Licence."
The link is too long, but it's on the TV Licensing website under the FAQ "Do I need a TV Licence if I only watch programmes online?"