Let me rephrase :
- You have all your equipment on the same LAN, with different subnet in it (so not using VLAN ...?) --> yes
- You have 1 subnet range, 192.168.0.0/24, on this LAN, containing VMs (let call them Group1) --> yes
- You have 1 router (let's call it Router1) within the 192.168.0.0/24 subnet range which give you access to Internet (IP 192.168.0.254). This is the "Common Internet Access" Router, right ? --> yes
- You have 1 subnet range, 172.19.0.0/24 (or something else very similar) containing server that your "Group1" VMs must connect. --> not excactly... the servers behind router2 have the same adresses like our local network. Don't ask why this was made up by some verry smart poeple that did not thougt about the possiblity that the adress range 192.168.0.0 could be verry common in use by the customers.
- You have 1 routeur (let's call it Router2) within the 172.19.0.0/24 subnet (IP 172.19.0.254). This is the "Private Cloud" router, right ? --> yes
One point I'm misunderstood with : what do you call "the router has also the 192.168.0.0/24" ? Do you have 2 differents subnet with the same IP range ? --> see above :-)
If that's the case, you need to use NAT while routing your Group1 VMs to your server within 172.19.0.0/24. No kind of problem, just some headache configuring your routingbox 
Well that could be done probably... and the more I think about that crap it seems to be the only relaiable solution.
On the other hand when the VMs could send the traffic to the 172-network it would be a faster way without natting... Not quite a fine solution but it'll suit it's purpose...