Well, I'm not in the UK, and we do have HIV testing as mandatory antenatal care here.
However some people don't present for any antenatal care here at all, and just arrive at the hospital untested to deliver, they have to be treated as if they were HIV/ HepC positive as testing would take too long.
historically, we have had a lot of African women coming to get their visas for the EU, by giving birth here, and until a loophole was closed we did have HIV HepC positive people giving birth in hospitals and leaving before their results came in. It was hugely expensive, as everyone of those mothers had to have barrier nursing.
I'm just discussing the possible risks. I'm a scientist and deal with risk. Maybe you find the risk too small, and so find it hilarious, but it doesn't change the fact that there is a risk. The fact that testing isn't mandatory in the UK is news to me. I think mandatory testing a good idea.
I never swim in public pools btw, they're even filthier than the sea, and fish fuck in that! 
Back to the OP.
I defend the mothers rights to breastfed where she likes, but think there would be more comfortable places for her to do so, without the risk of her babe and herself getting cold.
Also, if there's a no eating or drinking allowed in the pool, then I think the mother should have followed those rules. Breast feeding doesn't make you "special" and not obliged to follow the rules.