Don’t worry, this is a common problem when connecting two organisations together, or two networks that have been configured in isolation.
On the WAN routers you can NAT the IP addresses of the remote site, so that site A thinks the other site (Site B) has a different IP address schema, and Site B thinks that Site A has a different IP address schema. This can be done on a network basis, so you don’t need to know all the individual host addresses. If it is just one host one one site accessing one server on the other, then it is even easier.
You don’t need to use VPN connections or anything like that. By the way, they will work on the inside, it is not a common use, but it does work with the correct config.
If you are using directory services, then you may need to play with these to get the correct addresses, but if services are accessed by IP address, then you just need the NAT.
The obvious solution is to re-address one of the sites, but I guess that is politically difficult !
If you need more help in actually achieving this, then post more here, but you may find it better to hire a local consultant who can do the work for you, and will have better visibility of the entire networks.