I'm from the UK and I will start work here in Poland in September. I called the local tax office and they said I need to go to Gdańsk to register myself but I won't be taxed by the Polish Government.
That's the 183-day residency rule. However - to avoid any potential hassle with the Polish tax office, I'd advise you to simply go through the Polish tax system - it makes life much easier and you won't fall foul of any tax law in the process.
In general though, if you're working here until June/July, then you want to go through the Polish, not the British system.