I didn't love it to be honest, but also I wasn't in Panama City, but we went most weekends.
If your DS is going to school, have a look at both Balboa Acadamy and Kings College. Both are international schools in a lovely green area called Clayton. Kings only opened last year and is British run with British teachers, so your DS could start there with no Spanish at all. I met the Head when we looked at it for my DD and she seemed great.
You will be able to find an English speaking nanny by asking around, but likely to be someone who just likes children and needs a job, as opposed to a 'nanny'. We had a lovely lady who my DD still talks about!
Will all school fees etc be paid? They are very expensive so make sure it's covered. And housing. Probably looking about $2000 a month for rent.
There are nice parks, but the humidity can be a killer for little ones. They don't really have activities for children, like a soft play for example, but once you meet people you can get together that way.
I would try to improve your Spanish before you go if at all possible.
There are lots of Americans living there and people from other Latino countries. Didn't meet anyone from the UK I don't think, but then again DD wasn 't in school.
Good hospitals and Dr's.
If you do get the job, PM me and I will give you some specific details of things like the English speaking hairdresser and good supermarkets etc!