My first wood wheel got pee soaked and hadn't been plastikoted
I did give it a good scrub under a hot tap and that worked fine, but it wasn't the one with the cork surface. I would stick it back in for now - and get another wheel that is easily wipe clean, then when you've replaced it, clean up the old one again and keep it for out of cage time/playpen time/emergencies.
They can be got much cheaper than £20! Zooplus sell the wood wheel for about £11(although there would be postage on top) or you could get the plastic Trixie wheel 28cm from Amazon for about £10.
If you get another wood wheel and give it two coats of plastikote, it will be fine. As Cypher says, most hams do pee in their wheels. I found our Syrians do occasionally, but not regularly.
At the moment, with the peeing, you have to be careful not to get into an "itch scratch" cycle. They sometimes pee on everything to scent mark it if there has been a lot of cleaning and their scent has been removed. Then you clean it up - and if you clean everything it makes them want to pee on it more.
So this is a case of time and patience and trying a few things. Do you have a litter tray? You could try putting a litter tray in with Chinchilla sand in it - somewhere near the house. Is there a corner of the cage where your hamster has pee'd (as well as everywhere else?!) - if so - put the litter tray there.
When they are babies also they haven't quite learned control habits. So they sometimes do pee in the nest. That usually stops fairly quickly - especially when they realise their nest has gone as they are very particular about their nest being messed with. If you have to take the nest out then try and leave a little bit of the old nest behind, that is dry - even if slightly whiffy - and let them rebuild it - so it still smells a bit familiar.
Avoid doing full cage cleans and it's not necessary to do them weekly. You can spot clean mostly and do partial cleans. So spot cleaning is removing the odd handful of soiled substrate and replacing it with a new handful - but try not to do it every day - they do get upset about cages being messed with. With a partial clean - when you need to change the substrate, just do the substrate and don't clean anything else. Then another week you can do the wheel and another week any toys etc - that means something always smells familiar so they don't get anxious and want to pee on everything to reclaim it.
Another thing that can make a big difference is actually the layout of the cage and the type of house they nest in. If it's a small plastic house that's solid underneath it will get stinky - they do much better with a larger house that is open underneath and sat on top of the substrate. That makes them able to have more normal behaviours, so they can bury hoards under the nestand burrow down a bit - and better ventilation.
And they're happier and less likely to pee in the nest - also it means they can stay at floor level when gong to the toilet.
Some cages like the Rosewood pico cage - the house is right at the top - along way for a hamster to go all the way down to floor level to pee on the substrate!
If you post a photo of the cage, it may just need some tweaking to help your ham get normal behaviours sorted.
Have to say though that this can be an issue with wood wheels - even if plastikoted - I have found it fine if plastikoted - but some people find they still get pee soaked - depending on the hamster.
I have the Silent Runner wheel - the 12" one - which is a great wheel and is silent - and plastic - but it is expensive. The 28cm trixie wheel is popular and cheap - usually not noisy but isn't silent.
This is the silent runner wheel. I have mine attached to the cage bars to allow for plenty of substrate underneath it. But the bar attachment kit costs another £5.
https://sugarglidershop.co.uk/produc...nner-12-wheel/
Trixie 28cm plastic wheel - not silent - but most people don't find them noisy - although you could be unlucky and get a rattly one.
https://www.amazon.co.uk/Trixie-Exer...xie+28cm+wheel
Zooplus Trixie wood wheel - maybe the one you have - needs two coats of plastikote on the wood and cork running surface. And that adds another £4 for the Plastikote.
https://www.zooplus.co.uk/shop/small..._wheels/609617
The silent runner is expensive, but I think it's a good investment. It is totally silent and spins really well and easily. It is also the only wheel I've had that hasn't broken! Had mine about two and a half years or more and it's on it's second hamster!
I really like it attached to the bars as it makes life easier. Depends on your cage though - the silent runner is 35cm tall on its stand - wouldn't fit in the Hamster Heaven for example.