I agree that it is probably too close the fence. You must plant trees for what they will grow into! And you need to be able to get around them to prune. I would move it while you can.
I totally disagree about letting it grow without a trim. The first thing I did when I planted a tree was prune it or had someone else do so. And I planted, or had planted, thousands. Why people are so afraid to prune amazes me. The trees cannot do it themselves you know?!
Pruning does several things. First of all you can resolve things like crossing branches on small trees with hand pruners that will grow to be major chainsaw problems. Second, pruning top growth stimulates root growth and reduces the trees need to commit fluid and nutrient resources to too much aspirating leaf growth. Finally, it lets you shape it into what you want and reduces the "sail" surface that has to respond to the wind load on it. That's part of your problem now.
It is controversial where I am but in N California where I practiced, young trees got at least one sturdy full height tree stake to get them a start. Sometimes they got two placed on the windward and leeward side. Or even wooden guy wires anchored to wooden spikes for larger caliper trees to hold it from 3-4 directions.
Buy adjustable tree ties or get the simple ones cut from old tires with wire on the end. Make a figure eight between the tree and stake. Replace or adjust often.
As for when to prune next? I would wait for the leaves to drop. If you live in a warm winter climate, you can fall prune. If winters are harsh and/or early I would wait until spring to see what cold damage has been suffered. If you wait until spring it is sometimes easier to see the buds so you know what to prune back to.
And please don't make your trees scream or hurt yourself. Nothing stings like a dull pruning saw! Use nice SHARP pruners and saws! Dip them in bleach between cuts if you suspect any kind of fungus.
Every tree owner should get a good book on pruning. Morrow used to publish a little one that was almost as good as any for general guidance I ever ran across. It is out of print and $30/used seems a bit much but it is called "How to Prune Almost Anything". Ask your REAL, not box store, nursery for recs too. Or visit the library and copy pages for trees that you have.
Where are you by the way? You might want to update your displayed profile with ss number, exact home address, bank account and routing number, passport number and birthdate. Or I guess knowing where you are regionally or city and state would be as helpful to us on this forum in answering yard care questions.