Unfortunately for you - it depends. Anyone who tells you otherwise is lying or mis-informed, so beware
Almost all lower-end laptops (budget range you are looking at), are optimized for battery life, not the VERY computationally demanding task of real-time video encoding. A gaming-oriented laptop would probably be a good start to give you system capability headroom (ie avoid running close to limit and bottlenecking/crashing/freezing computer)
And it depends on how sophisticated you want your stream to be. A £500 could be fine with an optimized Operating System and OBS configuration, and realistic expectations. But... Start adding CPU intensive features like noise suppression, chroma-keying (green screen effects), poorly written plugins like streamelements, etc and system requirements can go up quickly (but it depends).
And how long do you want this system to last? Most live-stream services today accept H.264 input as H.265 is a licensing mess. AV1 will likely supplant H.264 in the coming years, but only some of the latest GPUs include a hardware AV1 encoder to enable CPU offload (and maybe only latest Intel ARC mobile GPUs?? I believe nVidia RTX 4xxx series may include AV1 encoder as well .. desktop models only at the moment)
If/when 4K streaming becomes the norm in a few years, will you want this system to handle that as well? or will you be ok buying a newer system at that point.
With all that that said, beware quality compromises on consumer (vs business) grade laptops.
- so read performance reviews and be attentive to thermal throttling and what, if anything can be done on a specific model to overcome such
- Assuming a desire for a multi-year life, with latest Windows desktop OSes, I'd advise against anything with less than 16GB RAM.
- And be thoughtful about your audio and video inputs, and the laptop chipset so as to avoid a USB Root Hub bottleneck.
Personally I'd prefer an AMD CPU, but their lack of overall system integration is problematic. So, I'll be waiting for a USB4/TB4 Intel Raptor Lake HX workstation laptop with hopefully a PCIe v5 NVMe SSD. But such a system will cost way more (4X+?) than £500, but such a laptop would also be complete overkill for what you are asking (and possibly not released/available for 6 months).
Some thigs to consider.
If your budget is key driver, then I'd recommend considering looking at an CPU with an integrated GPU (low performance, but possibly adequate for 1080p streaming ? For example, is 30fps ok, or are you expecting 60fps stream/recording?