I have about a third of an acre that was pretty full of self-planted trees, bushes, etc.
I have given away quite a bit of wood on my local Freegle group,
https://www.ilovefreegle.org/ ,
not just logs but also thinner stuff. Where you are there may well be people with open fires.
I have a shredder and I have shredded lots of the thinnest stuff (up to 40mm), some I have used on the garden and some I have given away on Freegle.
I do have an incinerator (a wire mesh box) in order to contain the fire like JohnD says, but I did not use it much and so I put it away.
What I do use is a large double wooden compost bin, rather like this
https://www.amazon.co.uk/Classic-Double-Twin-Wooden-Compost/dp/B0016N6OTG
which I got for nothing on Freegle! However you can build something with pallets or scrap wood.
I have a number of Dalek compost bins but stuff took quite a long time to break down in them, they are not that big, 220 or 330 litres. Each part of the wooden compost bin holds c. a cubic metre and stuff in there gets a lot hotter (>50C easily and 65C at times) and breaks down a lot faster. I filled both parts with loads of shreddings last June time, I merged them when they had broken down enough and by November the compost was broken down enough to (mostly) go through a 1" sieve. I used quite a bit of that compost in March when I planted some trees and a hedge and it looked good, well broken down.
Most of my trees (well the ones I cut back / down) are laurel, but I have a fair bit of holly and some yew. It all goes in the compost bin.