That's apple,..you did good by removing dead branches, a little more cutting out to open it
up a little can be done. Time will tell if it sets fruits,..how far away are any other ones, if no others within a block or so you might be apple to cut some branches from a crab apple somewhere and set it here in a bucket of water.

There are two trees beside each other. They look identical - does that matter?

They sure look the same..since it might have been planted by bird dropping..[seedling] it could be suckers from the same tree and therefore will not help to pollinate..most likely there is something else to pollinate with,..in a couple of weeks you'll know.

How will I know in a couple of weeks? Sorry, I know nothing of fruit trees.

First those buds will open. Then, after the petals drop, you should start to see tiny embryo apples forming and growing if the flowers have been pollinated. Keep a regular ( i.e. almost daily) watch on your trees so you can learn the sequence of its growth through the year.

If I understand correctly, then they have about another week to be pollinated? So today I should rush out and find an apple tree somewhere? lol

Maybe, maybe not. Those definitely look like Malus domestica flowers to me. If the tree will produce fruit, you should be able to see some of the old shriveled flowers starting to swell at the base. Many times, those old trees become neglected and, while still alive, stop producing fruit. Pruning out the dead/diseased wood and a few of the big branches in the middle to open the tree up can reinvigorate the tree and allow it to start producing again. Careful though, apples can be tip bearing or spur bearing. New apple owners can prune all of the spindly twiggy spurs off thinking they were unimportant, only to find they pruned off all of the apple bearing wood. This is what to look for on your tree.

I will go down and look straight away

No, I have nothing like that. There are a couple of flowers, but none shrivelled and the rest are still buds.

>>If I understand correctly, then they have about another week to be pollinated?<<

Yes

Update. There are apples on the tree now; they are very small and green and I have no idea what they are. They are the size of limes. Does anyone have a clue?

Thank you for any help.

That might pretty well be a seedling as suspected...see how it taste when ripe and if useable, [when seeds are brown]. Is it growing on the north side of the bush?

Forgive my ignorance, but what do you mean by a 'seedling'? Yes, they are on the north side.

Grown from a seed, [bird poop]. You have a one of a kind apple and can name it yourself.

Nobody would plant a apple tree on the edge of a north side bush.

Thanks for the info. The property was subdivided about 20 years ago so the tree would have been in the middle of the property previous to that happening.

But still on the north side of a mature bush with over 20 years old trees, older then the apple tree, perfect for birds to roost and poop.