If we look at the stats from Bitnodes we can see that there are total of 8861 nodes and I am not sure if they are all full nodes or some of them are just watch nodes.

https://en.bitcoin.it/wiki/Clearing_Up_Misconceptions_About_Full_Nodes#Very_roughly_estimating_the_total_node_count

This number comes from sites like this and it measured by trying to connect to every nodes on their open ports.

Problem is, not all nodes actually have open ports that can be probed. Either because they are behind firewalls or because their users have configured them to not listen for connections.

Nobody knows exactly how many full nodes there are, since many people don't know how to forward ports behind a firewall, and bandwidth can be costly, its quite likely that the number of nodes with closed ports is at least another several thousand.

...

Luke's script
Luke-jr has a script which attempts to count all full nodes: http://luke.dashjr.org/programs/bitcoin/files/charts/software.html