Oh, this is annoying. Not that I would, but what happens if I refused you entry to the property.
You can't refuse us entry. Fire and rescue services actually have stronger powers of entry than the police do.

However, it shouldn't need to come to that. If you are going to be burning anything at all in your back garden that may well result in a reasonably large volume of smoke then I'd suggest giving your local fire station or service HQ a quick call ( don't dial 999!! ) and just inform them that you intend to carried out controlled burning at your address. They will probably ask you to give an estimate of times that the fire will be burning and maybe request that once you have finished that you give them a call back to let them know.
Otherwise as Eddie says, you may well find 2 x fire appliances at your door. What can sometimes happen is an individual with good intentions may well see the smoke from a few streets away and call us. If the individual on the phone states they don't know what is on fire but there's lots of smoke, our control will generally mobilise two appliances ( sometimes one, depends on the service area PDA policy ) to investigate. If the householder is present and states its controlled burning then we generally wont extinguish it, unless it is near to adjacent property or overhead power cables or if it consists of vast quantities of acrid black smoke billowing everywhere. But a quick phone call will prevent the arrival of two fire crews in the first place so for the sake of a few minutes on the phone, it is worth doing.
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