The sizing is the frame size. Wheel sizes are standard for any particular type of bike.
You're about the same height as me and you probably want a medium or large frame bike, depending on the sizing of the range. If your proportions are typically female (longer legs/shorter torso) you'll probably be more comfortable on a bike designed for women, but it might not be a huge factor.
If you're going to be mainly using cycle paths/road/other hard surfaces you probably want a hybrid bike. Road bikes can be difficult to get on with if you're not a confident cyclists and mountain bikes add extra weight and complexity that aren't needed if you're not riding on rough terrain.
Second the suggestion to look at a proper bike shop, or see if you can get a second hand one. Or borrow a bike to try one.
There does seem to be an emerging market in businesses selling second hand bikes (that they've bought from people in a 'we buy any bike' sort of arrangement) which might be worth investigating as your money will go further. Eg
https://www.cycleexchange.co.uk/
Even Halfords are in on it
https://www.halfords.com/bikes/second-hand-bikes/
(Halfords aren't great for bikes - their own brand ones are pretty terrible and their knowledge and quality of assembly is very hit and miss but they have some second hand Boardmans that might be worth a look).