Now for the nerd answer!
Assuming taking from point where has been dialled in on MCP and register has been passed to the aircraft transponder:
Slowest Mode S ENHS radar feeding the TMA is 8 second scan rate. So if pilot changes just after last detection, worst case 8 seconds before radar downlinks the SFL register.
Radar allows for up to 2 seconds between detection and output to allow for the longer time required by the primary radar to process its data and the combined data to be output.
Data to get to centre is measured in milliseconds, so will discount.
Data processing works on 4 second cycles, so worst delay there is 4 seconds.
Maximum time to present to controller should be 8+2+4 = 14 seconds. In practise is much less as multiple radar sources extracting data.
RS