Which model hub did you buy? (The hub model should be on the bottom of the hub).
A lock will be either Zwave or Zigbee, not both. Just look for the logo on the box.
As @withay mentioned, the Schlage Connect comes in both a zigbee and a Z wave model. So if you bought the Z wave model (it says Z wave on the box) then you would connect using Zwave.
Zwave uses different frequencies in different countries
However, I am concerned because you said you live in the UK. Z wave operates on different regional frequencies, just like mobile phones do. The reason is because ambulance services in different countries operate on different frequencies and they want to make sure that a zwave device will not interfere with those.
The Schlage connect lock is made for the US market, and consequently uses the US Z wave frequency.
The regional Zwave frequency is assigned at the time of manufacturer and cannot be changed afterwards. And just as with mobile phones, the Z wave lock and a Z wave hub must be using exactly the same frequency or they cannot communicate with each other.
If you bought your smartthings hub in the UK, it is most likely a UK model operating on the UK zwave frequency. That means it can only talk to other Zwave devices using that same frequency. It cannot talk to devices using the US Z wave frequency. That’s why the hub model number matters. It will tell us which zwave frequency it is using.
(BTW, wink also operates only on the US Z wave frequency. Smartthings, however, makes hubs on multiple different Z wave frequencies. So you buy the UK model if you live in the UK, you buy the Korean model if you live in Korea, you buy the US model if you live in the US. But again, that means the hub can only talk to zwave locks or sensors of the same regional zwave frequency.)
All of which is to say if you have a UK hub and a US lock, they cannot talk to each other using zwave.
US hub and US lock: Schlage-specific issues
If instead you bought the US model hub, it may not be legal to operate in the UK, but that’s a separate question. The US model of the smartthings hub should be able to communicate with a Schlage connect Z wave lock. However, there is a trick to it because Schlage has A very precise pairing sequence which depends in part on the physical location of the lock relative to the hub.
There is a community FAQ which details how to add a Schlage Z wave lock to your smartthings network. Again, though, this will only work with a US model hub.
FAQ: Pairing Schlage Z-Wave Locks
If the logo on the box is Zwave, then pair it using the Z wave option. Not Zigbee.
Check the model number of your hub. If it is a UK model, it will not be able to communicate with a Schlage connect zwave lock because they are using different regional frequencies for zwave. There’s nothing you can do about this. Only the US model of the hub can communicate with a Schlage Z wave lock.
If you do have a US hub model, then follow the instructions in the community FAQ at the link above.