As others said, can you test your speed with only your work vpn?
What is your work vpn? is it Cisco anyconnect?
When using double vpn, generally mtu will be a problem and cause serious lagging. Do you have any way to adjust your work vpn mtu? You can also try adjust your computer's mtu.
Check out this one if you have a Windows How to connect Cisco Anyconnect when using VPN on the router (double vpn)