... be patient, either they will or they wont.
This is good advice. You're dealing with trees, so you're talking about seasons, not days. Figs have at least two years' worth of fruit on them, so you only want to pick the ones that are ripening now. There might not be any left on your tree, the birds will be up a lot earlier than you.
Figs are one of the most delicate fruits which is partly why you see them less often than things like apples and pears. They don't keep well, even refrigerated, unless dried or preserved -- but the preserves are delicious! There's some advice about checking for ripeness
here. You're looking for something slightly soft which smells nice. It's not rocket science.
The tree might drop its fruit before they're ripe if it isn't happy, if it does that you can try to ripen them at home but be prepared for disappointment. You're going to have to learn about your tree. There are many varieties, they produce fruit of widely different colouration and ripen at different times. I don't have a clue what variety my figs are.
I don't think they will ripen once picked though.
They will if they aren't too green when picked.