Are the routers configured either with a default route pointing to 192.168.1.1, or at least with a route toward 10.10.10.0/24 pointing to 192.168.1.1? If not then these routers have no idea whether 10.10.10.1 exists, and if so, how to reach it.
Okay. Can you confirm that these two routers can ping their gateway 192.168.1.1?
Nonetheless, you have just said that there is no other device connected to the switch except these routers, and the routers are connected to ports placed in VLAN10. That means that there is no switchport in VLAN1 that would currently be up. As a result, the switch has no physical port capable of delivering frames to and from VLAN1. As a result, I suspect that the interface Vlan1 on your switch is currently up, protocol down (check the show ip interface brief closely). That would explain why you cannot ping 10.10.10.1 - because the switch keeps the SVI for VLAN1 down if it has no physical switchports alive in VLAN1.
Based on the information given, it is very likely that the scenario described by Peter is happening. If there is no interface in up/up state which is a member of vlan1's SVI, then interface vlan1 will be in up/down state, preventing a ping from either PC to the 10.10.10.1 on vlan1.
Can you "show ip interface brief" and provide us with the results to verify? I am guessing this is just a testing environment. Do you need to reach 10.10.10.1 on vlan1 just for management? If so, you could use a loopback interface and assign 10.10.10.1.