Many years ago I had neighbours that constantly had a bonfire. We'd just moved in, hadn't met them yet, and it was awful. Because of the layout of the surrounding buildings the smoke would sort of collect in my garden and hang around. You could see it in the air. I couldn't have windows open or hang washing out. I was working 12 hr shifts at the time, so would often hang laundry out the night before and then come home to discover it stinking and have to be washed again. So I get it, I really do.
One day I came home early to yet another bonfire and a load of washing that I had washed 3 times stinking of smoke. They happened to be outside as I went outside to pick it in and wash it for the 4th time. They cheerily shouted over 'afternoon! How are you?' I lost it, I shouted at them that they were unsociable and selfish, that I was constantly washing over and over again, couldn't open my windows or sit in the garden.
They were...a little taken aback, shall we say!
We lived in that house for 12 years, there ware only 6 houses in the little hamlet, and we all knew each other. I'd say it took a good 4 years for us to get over my outburst and have a neighbourly relationship. It gradually became friendly. They were lovely people who genuinely didn't realise they were causing a problem. Many times over the years I wished I'd gone and knocked on the door and told them I was having a problem, and asked them if they'd let me know when they intended to burn stuff as I wouldn't be home to pick washing in. They were so nice, they would have had no problem confining their bonfires to Wednesday's or whatever and probably would have offered to pick my washing in themselves if necessary!
So my advice is don't contact the council in the first instance. You'll inevitably fall out with them and you do have to live next door to them, after all! I'd pop around, possibly with a 'new home' card so the visit isn't all about complaining, and politely let them know that the smoke is causing you problems. If you have a green waste bin that has space in it, I'd offer them to use yours too to help them shift the stuff they're clearing so they don't have to burn it. We have to pay for our green waste collections, but in your situation id be happy to trade that off against a smoke free garden!
If they don't respond to you trying to meet them half way and carry on burning all the time, then at that point I'd go to the council.