How will a router with NAT disabled behave?

Solution 1:

With NAT enabled, the router will modify the IP header in outgoing packets so that the source address matches your Internet public address (and vice versa for incoming packets).

If you disable NAT, it won't do that anymore. So, basically, you will be sending IP packets with private source IP address (e.g. 192.168.x.y) on the Internet, which of course will automatically get rejected by your ISP.

Solution 2:

NAT is what lets you use internal-only IP subnets, e.g., 192.168.0.*. While routing, the router will substitute its address for the internal-only address. Unless you have a subnet of real IP addresses available to you, don't turn off NAT.