You should not use the same domain name for your public Internet domain and your Active Directory domain.
When you create an Active Directory domain, your Domain Controllers become DNS servers for a zone with the same name as your AD domain (in your case, "example2.uk"); this means they'll become authoritative for this DNS zone, thus they will never query public Internet DNS servers for its contents; they will just answer any query with what they know about the zone. This is called "split-horizon DNS".
In this scenario, if you want to resolve public names such as "www.example2.uk" or "mail.example2.uk" from inside your network, you will need to manually create records in your internal DNS zone and have them point to the real servers hosting the services; this causes some management overhead, but it can be done... except for the root of the zone.
In an Active Directory domain, the name of the domain (such as "example2.uk") must be mapped to Domain Controllers; this is automatically managed by DCs: each DC will automatically register an empty A record pointing to its IP address. If you look at the DNS zone in your environment, you'll easily notice those records.
This means you can't map the name of the domain to the IP address of a web server.
