I wouldn't worry too much. If the payment came through as PAYPAL then they are probably using PayPal as the payment processor even if it is only accepting debit card transactions. That gateway is implemented by PayPal (even if it's just literally copy and paste code) but they could just be spoofing the bit of text that shows up in your bank. It takes a LOT of work to create a dodgy card transaction processor from scratch which is why even scammers will just implement PayPal because it's faster.
Oddly the site's footer has the Stripe logo instead in which case I'm not sure why it would come through as PAYPAL on your statement, as Stripe definitely looks different there (I use both).
The number will be VOIP redirected to some number wherever and the address is probably a virtual office in London, although even that seems dubious.
The photos are obviously old or appropriated, probably the latter.
Regardless of where they are actually based, they're still subject selling in the UK to distance selling regulations. You have a right to cancel within 14 days and return for any reason within 30 days of the item arriving. As nothing has been despatched you have a right to cancel the transaction and get a refund anyway. It's just getting through to them that is going to be the hard part.
I'm amazed Lime even found the company registration details. You could try contacting company's house if they're not acting in the proper manner for a UK company with the potential for them getting struck off the company register due to fraud. But they will just spin up other some other guise. The suggestion here is that, if they are actually registered, this is just some joker/chancer trying to run a crappy games retail outlet in this country.
If you got a PayPal receipt from PayPal (regardless of whether you signed into your PayPal account or not, PayPal still acts as the payment processor for the card details) you could try contacting them.
If that doesn't work or you don't want to wait for the bank to chargeback the transaction (which is still possible on VISA network, absolutely, whether a debit OR credit card, it doesn't matter) you could just go straight to trading standards. You've made every reasonable attempt to contact them and haven't got a reply.
They seem to be using a support ticket system as well, maybe try this form instead of emailing? They probably aren't monitoring that inbox.
https://citygamez.com/submit-ticket-2/This seems more a case of just a crappy and poorly thought out business set up in a weird way to be honest, hopefully you'll get your item or your money back my eventually. If you were super worried about it I'd cancel the card and get a new one because they could (theoretically) record those details and keep/sell them. I really wouldn't rely on this site to comply with the PCI requirements for storing and processing credit card information, so hopefully PayPal did actually process it in which case it's safe. What did the form look like when you placed your order?
It's good you posted about it too as within a few days Google will pick this up, list it in the search results probably near the top and discourage anyone else from ordering from them. A crappy business can refuse to set up any kind of reviews/social media presence for a site like this, to avoid doubt, but you can't stop people talking about it on some other platform with good search visibility. As their site is so poorly written we might even become the no.1 result instead of them
Edit: That address is the offices for "Registered Address Ltd"
how deep is this rabbit hole
And how does Companie's House allow these onto the company register with no staff actually present in the United Kingdom
Edit 2: Wow
RegisteredAddress.co.uk is pleased to announce a new upmarket virtual office address:
South Molton Street, Mayfair, London, W1K.
This office offers a mailing address in the premier luxury retail district of central London.
The office interior is a standard office layout. Located on the second floor, it includes a meeting room, desk spaces and open-plan office space for corporate users. The meeting room can accommodate groups of up to 6 people. At ground level there is a boutique.
This office establishes the client as having a prestigious presence in a very expensive area of central London. The area has name-recognition for both UK and foreign residents, so it will add cachet to your new or existing business if you use it as your registered address, director's address or business address.
So basically this seems to be a legitimate service that allows international businesspeople to form a business in the UK although of course this means there is a pretty difficult path to take to hold them to scrutiny.
https://www.registeredaddress.co.uk/officeservicesIt
might work as a freight forwarding / drop shipping business but I wouldn't get your hopes up this item is going to reach you any time soon, which is, well, bold considering the company goal is to "deliver games
instantly" (emphasis theirs). How they intend to do that with nobody inside this country is anyone's guess.