It does not sound like you have a device configured to route between these VLANs.
All you have laid out is layer 2 switching, but have not talked about what (if anything is setup to do routing).
You would have to have a single device connected to all VLANs (either an MX or switch with L3 routing turned on).
You cannot forward traffic between VLANs without a routing device connected to all VLANs
That routing device would have an IP address in each VLAN, which would use as a default gateway on that VLAN.
Also as @PhilipDAth says, all of those VLANs would have to be defined in each switch.
This is because the non-Meraki switches are not learning the VLAN from the Meraki network.
Is there a reason you have to cascade the switches, rather than have both Switch 2 and 3 connect directly to Switch 1?
What model is Switch 1? Is it Layer 3 capable, or do you have an MX?