What PC upgrades would give me the biggest boost for £300?

You could look for a used RTX 3060 Ti at roughly £200 and buy used RAM as well. It would improve gaming and GPU-accelerated editing, but make sure the power supply has enough capacity and the right connectors before buying it.

The easiest first step is adding another 8GB DDR4 stick with matching speed and timings so you can run 16GB in dual-channel mode. That should help general performance. If the hard drive is being used for games or applications, adding a SATA SSD would also make loading and editing workflows feel much faster. A used RTX card with 8GB of VRAM could be considered later, but check your power supply and case compatibility first.

For DaVinci Resolve, more RAM and a faster processor may be more useful than a small graphics card upgrade. A used Ryzen 5 5600 with a compatible B450 or B550 motherboard, plus 32GB of DDR4, could fit close to the budget if you find good second-hand prices. A Ryzen 5 3600 would leave more money for memory, though it would be a smaller CPU upgrade.

A motherboard bundle containing an Intel Core i5-12400F and 16GB of DDR4 might be available for around £300. That would give you a substantial platform upgrade, although the best choice depends on which games and resolutions you play. Your existing GTX 1660 could be reused for now.