The BBC supposed to be nonprofit
No, not in USA or Charity sense.
There is BBC Enterprises selling stuff in UK and BBC Worldwide (I think) non-UK.
They are not* supposed to advertise on BBC Radio or TV *IN THE UK*. So while all of U.K. (and Ireland) can receive BBC World TV (Free, with Adverts) via satellite, it's on a DIFFERENT satellite to the one all the regular UK TV is on (Sky & Freesat Ghetto TV), some people have a multifeed satellite dish, but this is MUCH rarer in UK or Ireland than Mainland Europe or USA.
The BBC is "non-profit" in the sense of too many managers, inefficient and always wanting to do more than what they have income to support. They ARE a Public Service Broadcaster (so in a way is UK Ch4, but it has adverts). UK ITV and UK C5 are purely commercial.
The BBC is also funded by the Licence Fee, essentially really a tax, not a subscription, as you pay it if you watch ANY UK TV. In UK you used to be able to get a TV licence exemption if you proved you only watched "Foreign TV". Such Exemption never applied in Ireland as the TV licence (Tax) started 6 months before Irish TV and might have been 7 or 8 years earlier (very many Irish people from 1950s could get BBC, then ITV. The Irish TV started 31st December 1961 at about 10pm, Irish TV DOES have adverts as well as money from Licence fee).
BBC & RTE (Ireland) are not PSB in the same sense as in USA. They both run mass market channels and make programs down to the lowest common denominator and chase ratings. RTE (due to 1/20th of licence fee income potential as Ireland is 1/20th population) is a Commercial broadcaster too. TnaG is the Irish Language PSB + Commercial Broadcaster. UTV-I and TV3 (once owned byy UK ITV) are the Irish equivalent of UK ITV and C5 commercial only broadcasters.
Then there are are the "multichannel" stations that before Terrestrial Digital were only on Cable and Satellite. They are almost all pay TV. The free ones get less than 0.5% viewing time.
[* BBC of course cross-advertises it's own channel content. In fairness I can't remember them promoting any of their content in the shops on UK TV and Radio, but then they don't need to]