Pro riders don't use £3000 bikes ... they cost a whole lot more than that.
If the cycling thing is just a fad or an aid to recovery then go for the £300 bike as more than likely it will be parked up against the back of the house and abandoned, before being taken to the tip when it's all rusty.
If you do go for the cheaper bike as a rule they come with knobbly off road tyres. Get them removed by the shop and have them fit some kind of commuter style tyre as they will be much easier to use on the road surface where the bike will spend most of it's time.
You can get better brakes, wheels, hubs ,sprockets, gears etc... this is how you get to the £1000 range very easily, but do you need it ?
My son often puts a bike together to go to the Alps in the summer he spends in the region of £5000 .. he ends up with a bike that you can barely cycle along the road as it s totally unsuitable. But put it on a mountain side and you have got the perfect tool for rocketing down what had been a ski course in the winter months. He then comes back from holiday and breaks it up and sells the parts !
Decide what you want from the bike and buy accordingly, don't be taken in by prices.