So how the admissions system works is - say you put 6 schools down. The council will take your application for each of the 6 schools and use its admissions criteria to determine whether you get a place. So you could, in theory, get a place in all 6 schools. Then it will look at your order of preference and give you your highest preference choice. So it's really important to put schools in true order of preference rather than because you think you have a better chance of getting into one.
Have a read of the council's admissions booklet (Enfield and/or Barnet) and they both explain this, probably a bit better than I have!
I am not sure how far you live from Walker and Ashmole but if it's within 0.5 miles I'd say you've got a good shot of a place. There are other schools which tend to be undersubscribed (like West Grove, Garfield) where you'd be pretty much guaranteed a place. Of course things change year on year. This website is quite handy as it tells you which schools you'd have got into over the last few years (although they haven't entered 2018 data yet): www.schoolcatchment.co.uk/
We chose Walker as it is closer. We wanted to be able to walk to school and to have friends that lived in the neighbouring streets. We also loved the location of the school and the community, "village" feel that it had. Plus it is an established school and has a history of really solid results.
That being said, I was really impressed when I went to see Ashmole, it looked like a great school. I know some people whose children are now in year 1 and they speak extremely highly of it.
I really think they are both good choices. And it's important to remember that all Enfield schools are good, so we are lucky to live in this area, even though it doesn't feel that way when you are in the thick of school applications and it is all so overwhelming and emotional (I know, I've been there!).