Sorry for the long, complicated message, it's very hard to give a precise answer without the specifics. I'm happy to explain anything you'd like to be clarified.
If it's their personal website :
Assuming you were using mobile data, they'll be able to see that someone who uses your mobile network accessed their website (assuming they have IP logging enabled). I'm not certain for every mobile network but Three and O2 IP addresses aren't localised so that won't immediately tell them your location. IP addresses are frequently swapped between phones on mobile networks as different devices connect and disconnect, which frustrates attempts to localise them. Various providers use data gathered from other sources to try to approximate the location of a specific address and looking up your current address on an 'IP geolocation' service will let you see how close the providers are (my IP showed a variety of places I haven't visited).
However, if they suspect you accessed their site and wanted to confirm it, there is a risk that your phone retains the same IP address for a longer period of time so, by using certain attacks (both ones you might see/have to act on and ones you don't), they could potentially send you a message (through email/SMS/instant messaging) which effectively causes your phone to contact their server again, linking that IP address (and the previous one if it's unchanged) to the original hit on their site.
Unless they're running an extremely unusual setup, the site won't be hosted (where your phone goes to retreive the site contents) on their laptop, as that would require the laptop to be running at all times. It will more likely be hosted on a server but, if they have IP logging, they will be able to retreive the log in the same fashion.
If it's a social media page :
You're almost certainly safe from the above (though they could have gone to specific lengths to still record hits, but this is unlikely and difficult to implement). However, if you were logged into your profile, improper privacy settings on certain sites may allow them to see that you've accessed their page.
In either case :
It's possible for certain authorities to get a list of the sites you've visited (though this is just the addresses, whatyou specifically look at is harder to access and the precise details of whether it's retained automatically for everyone or only recorded after a request has been filed isn't publicly disclosed).