Thank you for the detailed response
There is generally 3 options. Standard tractor type, either with a sealed off deck or a mulch plug fitted. These are dead on, generally work better as a collector machine but good if you need both options. Start at £2k for a 36" machine new.
Front deck mowers, the 2 best manufacturers for these are Stiga and Husqvarna. The advantage of these is manoeuvrability and they leave a superb finish. Also good for under shrubs etc. Downside is they're not the most robust thing if you bump a tree, and they don't love long grass if you've been on holiday for 3 weeks you'll need to cut it 2 or 3 times. Start between 2.5 and 3k new.
Zero turns. An American design but I like them. Go for Ariens, they offer everything the dearer makes like Toro etc do but for less money. They are very fast, robust and exceptionally manoeuvrable. Downsides are the finish isn't as good as a front deck, and they can mark the grass till you get used to how to properly drive a skid steer. Some never do. £3k will get you a twin cylinder 42" new.
Used is a minefield, mowers aren't like cars, people don't trade in every 3 years, more tend to keep them till they start to give bother. Mower parts are expensive, every bit as expensive as car parts. As an example the average Husqvarna clutch is over £200 for the part alone, plus you'll need belts etc on top of that. For that reason I'd always advise new unless your reasonably mechanical and the machine checks out.