Model Runs on the Web

It would depend on the website you use... If you run software like GEMPAK or PCGRIDDS and generate the data on your own PC, you can start getting it after around 1440Z / 0140Z.
What time approximately do the 12z and 0z NAM & GFS model runs generally become available on the web for viewing ?

Most of the time, you can start getting model output 2.5-3 hours after initialiazation time. Your best bet is the NCEP/EMC website, which seems to be quite a bit faster than CoD, WXCASTER, or RAP/UCAR in updating.
I often find that RAP/UCAR is about one forecast step ahead of the NCEP model page. When NCEP is processing the 36hr FCST timestep, RAP/UCAR is on the 42hr FCST timestep.

I don't have my GEMPAK servers up and running since they tend to get way to hot (my server room, aka closet, get's up to 100-110F), but that usually comes in as the data is sent over NOAAport, or about the same speed as what the NWS gets it.