I believe is it wise to keep three separate subnets. That will avoid sending broadcast traffic between sites. Also, it will mean each site has its own DHCP, which they needs since they will have different default gateways for the LAN, etc.
NAT is, most likely, disabled for traffic through VPN tunnels - like today.
About the routing from the RV042: Today you probably have it set-up with a public IP on the WAN side, and with a WAN side "default route" to the IP of your cablemodem, or whatever your internet access is. With the new set-up, with the WAN side of the RV042 connected to the fiber (through some media converter or CPE), the WAN IP will be something like 192.168.x.1 and 2 and 3 (subnet mask 255.255.255.0), as you said above. That means that the 3 RV042 will share the same subnet on the WAN side, which means you will not need a "default rute"/"default gateway" on the WAN side for them to be able too "see each other" and bring up the VPN tunnels.
Once the tunnel is up, routing through it is usually 'automatic' - as today.
Assume you manage to disable NAT and tunnels on the RV042, which would mean it will work as a simple router. Assume now that a PC on LAN1 has a packet to send to LAN2. The PC sends it to its default gateway (unless to configure routes in each PC). The default gateway is the SSG-5, which has a route for LAN2 network 192.168.2.0 /24 pointing to the RV042. The RV042 receives the packet.
Now, if there is no VPN tunnel up, the RV042 in LAN1 will have no idea of where 192.168.2.0 /24 is located. It would need to have a static route sending network 192.168.2.0 /24 to next hop 192.168.X.2 .
This basically means that if you do not have tunnels, you will need static routes between sites.