Yes - double your budget. Editing 4k video on a £500 laptop is not going to be a smooth experience.
Some good deals do come up, there's been some stonking Lenovo Ideapad deals recently. Dell have had some decent Inspiron 5000 deals too with AMD Ryzen 4000 CPUs.
Although now expired I got an Ideapad 15" for £500 - 8c16t Ryzen 4800U, 16GB RAM, 512GB SSD.
AMD has bought some serious CPU power into affordable laptops.
don't see this 4800u AMD Ideapad 15" on there website. Was it a model being replaced