I think a simple Gigabit Layer 2 switch will do what you want and they are reasonably cheap these days. If you want to go really cheap then perhaps a 100Mbps switch with a 1Gbps uplink. You may need more than one switch to service this many ports.
Switches have different techniques for transferring data from port to port but in the worst case it will have enough buffer to queue packets while it waits for the transfer. In your case with 60 cameras and assuming 10Mbps (yours are probably less than this) you would have the 1Gbps link running at 600Mbps which is fine. There would be minimal transfer delays between the ports.
As far as i understand a swicth will momenterily connect port to port. So if the devices are limited to 100meg thats as fast as the date will go into the gigabit port.
The links are serial in nature and typically have queues even on the cheapest ones so your uplink should not be limited by any input. In other words, you would have multiple 100Mb links feeding into the egress queue of the uplink and the uplink port would be emptying the queue as fast as the uplink will allow (ie. 1Gbps).
Not knowing your budget, D-Link and Netgear both make very inexpensive 24-port L2 switches and may offer some basic management capabilities. HP is a good and reasonably priced commercial grade product but there are other good choices in this category.
Hope that helps.