Suppose I have a 500Mbps ISP connection and a wireless mesh WiFi network consisting of 2 APs. For the sake of argument, we'll assume that each APs Max Data rate is 1200Mbps.

What would the (theoretical) bandwidth to the internet be for devices connected to an AP that's a hop away from the router? Since mesh cuts the bandwidth in half for each hop, would the theoretical max be 600Mbps - 1200/2 (in which they could theoretically saturate the entire WAN connection)? Or would it be 250Mbps - 500Mbps / 2?

In reality L3 throughput over wireless rarely exceeds 60% of air interface rate. Which means that 1200Mbps radio will likely be able to sustain something like 700Mbps of data throughput. Meshing then halves this ... reduction might be even larger because then timing gets into the picture (air time congestion between different devices that have to push data one to another).

So in this simple case I would expect to see realistic data throughput in range of 250-350Mbps.
And that's in case when all wireless links are in ideal radio conditions (which likely means that meshing is not necessary in first place as station, connecting to mesh node, might be able to connect to anchoring AP and with decent speed). If wireless network is stretched, then "hidden node problem" might kick in (the mesh node would "hear" both anchor AP and station, these two would not "hear" each other and could thus not coordinate Tx attempts), which then further cripples performance of the whole network.