Not being able to get replies back would mean that setting up a new TCP connection would be impossible because you're not getting replies to the initial handshake. The TCP connections that were alive when the routes were dropped would time out pretty quickly if they weren't getting any replies. Whether UDP packets could get out is going to be down to how the connectivity was dropped. If it's just via BGP then there's a chance.
I guess it depends on the equipment, you would have to configure them to 'default route' non-local IPs to some upstream provider, but since router doesn't know how to return them it probably just whines. But if you're cutting off the country (and it did look like that from the Cloudflare page that someone in Syria configured the routers to stop advertising a network) then you probably wouldn't install a default route to a hop outside the country.
I'm not really routing savvy, but isn't it a little bit more like, the traffic could get out but no one could reply?