The catchment for Ashmole is currently about half mile, but I imagine that will decrease in the future, as they've said they'll give priority to children who attend the new Ashmole primary. Not sure what the catchment was for that though. It is an excellent school though.
Since Southgate got a new head a few years ago, it has dramatically improved, and I think the catchment is decreasing each year. However I think it remains around the 2 mile mark (by which I mean that was the furthest away a child lived in order to get in this academic year).
If you were towards the Trent Park end, you might get into East Barnet, which is also an excellent school.
Broomfield doesn't have a great reputation, but is supposed to be vastly improved, again since getting a new head.
Primaries in the area are all pretty good. It's really hard to get into Walker, Osidge and Eversley - loads of siblings every year. The new Ashmole primary will almost certainly have the same problem in years to come, but this is only its second year. West Grove is a vastly under-rated, but excellent school (my DDs went there). It's viewed as a last resort school by many, because it has a high turnover of pupils and a very high proportion of children with English as a second language. But the teaching is excellent as is the pastoral care.