You need an Amendment to Written Statement of Particulars to reflect the change - basically, an amended contract.
I work in HR and we have a few people who work some days of the week at home. Our system allows us to see the working hours that the employee does, and we could question it should there be no activity for a few hours. Most companies use email these days so if anyone needed you, you should be on hand to answer them. The company may provide you with a laptop or they would need to securitise your PC to ensure security is the same as it would be in the office. They would also need to look at the Health and Safety aspect - you are still under a contract by them so if you tripped on your laptop wire, they would need to ensure you can't sue them.
Working from home is ok, so long as you're not in a call centre of some kind where it's not practical.