I have (well had, the zip burst) a backpack from morrisons, it contained blastic crockery and cutlery.
I think the art of a picnic is to have as many treats as possible so home made quiche, smolked salmon, nice sausages cooked and cut up.
The next two things are people and location, I'm quite happy on my own but have done picnics for groups.
Location, well this is individual, not parked up in a lay by (I know you said you do not drive but I'm referring to a memorable one where my mother shouted at us that 'You will sit down, you will eat it and you will enjoy it').
So pate bread and a glass of wine in the back garden is fine. If you are anywhere near London then Kenwood house (postponed this year) have great concerts and you can hear the music in the grounds.
So youget music, but in the background so your toddler can run around, as it drops dark you get fire works and then, actually by then you will be carrying a todler and picnic so maybe not the perfect end.
If you can get to Oxford then Christchurch meadow is fab (particularly when drinking champagne and holding up score cards to the people punting).