Firstly, think about whether you want to run this as a company or whether you just want your product available on the shelves.
I had an idea, went to the expense of setting up a company, a friend did the branding, I got them made and thought a particular brand might bite my hand off to market it as their own. They didn't. I sell my product myself and do well but I hate running the company. It's a great product but I'm kind of accidentally dealing with corporate bullshit and ridiculous margins. I can't patent my idea because it's not physically new, it is a tweak on an everyday product.
If you just want to get the idea out there, pay for the patent, you don't even need the expense of a prototype and then approach the market leaders that retail "own brand" and let then manufacture/ take the risk for either sale royalties or a one off payment.
I started off saying "I'll get it made and see if it works" but the cost of getting one made means I may as well get 500. So I bought 500 and sold half and sent half out as marketing samples. And now I've ordered more stock to keep up with demand but hate that the risk is on me, the margins aren't big enough to take on an investor.
You only need to register with companies house once you start trading. The most important thing is the patent in your case (min 3k) If you want to sell it the idea you don't even need a name for it, if you want to sell the 'business' (branding, website, supplier, retail contacts etc) then the risk is all on you and bear in mind most retailers want a 70% mark up. So if the product costs you £1 to make, you sell it to retailers for £1.50 and they will rrp it at £2.55. Will your customers pay £2.55 or is it only really worth to them £1.50? Sorry that's not very clear but hope you get what I mean. Think of what people would pay, subtract 70%, then subtract the cost price and see if what's left is worth it for you. Also those retailers will want exclusivity for 1 year- that's a big risk too.
Getting it made is easy enough, I did mine through Ali baba but came close to getting ripped off a few times, by the time it was shipped and i paid import tax I would have been better off using a uk sourcing company. The quality is guaranteed and much easier to deal with. One I highly recommend is effectual sourcing- I did a tonne of research with different companies and they came out cheaper and easy to work with.
Hope that is useful, sorry if it is slightly negative, I'm projecting my own naiive self! Good luck, there is nothing better than seeing a physical product on the shelf which came from your brain!!