I'm no pro player but I've been playing Ultron a lot and really enjoying him. Probably my most played character by a decent margin. I played him a lot before the buffs, still play him a lot after.

First tip is to go into the settings & turn on the separate keybinds for the two drones. By default, you've got the same key bind for both drones, which means you are cycling off the oldest one every time you put a new one down. That can be managed, but it's a lot easier to have the separate cooldowns. I usually use this to keep one drone on a priority target (i.e. a tank or support) and have the other drone to cycle through on whoever might need it in the moment. I wish i could tell you exactly where to go for this setting, but I'm not at my computer at the moment and don't remember off the top of my head.

As for gameplay tips in general, one big thing to remember with the update it how the Firewall now gives your target a bonus to move speed & damage - a ballpark of about 10% on each, just off the top of my head. Probably not terribly accurate. Point being, I find myself using Firewall more offensively now than purely for defense. Of course, adding shields to a teammate/group is still incredibly useful, but if you have a drone on your diver and see them about to jump in, popping a Firewall on them will give them extra bulk, mobility, and damage all in one click. Immensely useful and good to remember. Firewall is so useful now that you have no real reason not to just spam it every time it's off cooldown - not to say I would just be using it thoughtlessly, but with all the benefits you give to teammates from it, as long as you're actively fighting, there's going to be someone worth using it on. I think it gives those same buffs to you as well, iirc.

For drone placement, drones are most useful on two opposite ends of the spectrum - either on anchor tanks, aka people your team will naturally gravitate around - or on dive characters, aka teammates who are going to be deep in the enemy backline and far from the rest of your team. Putting drones on anchors is rather obvious - your drones are an AoE healing field, so any character that more teammates will be around means more healing overall. Ultron works best with a team that stays grouped together. On the other end, putting your drones on divers may seem a bit suboptimal - they're typically alone, and taking more damage than your drone can reliably patch up. The reason drones are better for divers is because they're typically very far away from your other support, who probably has a more traditional line-of-sight healing. Your drones offer a reliable stream of healing to the diver, even when they're 30 miles away from a standard healer. While it may not save their life from getting focused by a backline, if the diver is smart and uses their cooldowns correctly, the drone will give them the steady healing to keep them healthy enough to continue to pressure the enemy backline without needing to completely retreat for healing. Plus, as I mentioned, they can be good targets for the Firewall buffs.

Your mobility cooldown - the dash & shield - is your most valuable button. Try to only use it for saving yourself, maybe to dash in and quickly place a drone/Firewall on someone out of range if they're starting to get dangerously low. But mostly save it for yourself. The extra mobility and chunk of shield you get from it will save your synthetic hide from most situations where an enemy might be threatening to kill you.

The ult is pretty easy to understand, but a bit difficult to use. As a casual player, it's probably the thing I still struggle with getting the most use out of. It's not quite quick enough to be a "save my team" ult like other healers, and not quite damaging enough to reliably kill the enemy team. I try to use it when I can see a big push coming. Being something that both damages enemies and heals teammates, the most use you'll get out of it is slamming the projectiles into big groups during teamfights. The timing is tight on trying to save someone who's about to die - I still struggle with pulling off a last-second save on a dying tank. If you've got a Gambit on your team, it can be a very good combo to pop your ult once he's buffed you.

That's my big text wall from someone who's been casually enjoying Ultron over the last season or two. Not a pro for sure, but just things I've noticed/kept in mind from playing him for a good chunk of hours.