I'll ping this a month late.
Rockwell is still king in the USA and Commonwealth countries. To be honest, it surprises me every day that this is the case since just judging based on price, performance, capability, MTTF, innovation, interoperability, and lead-time; Rockwell is garbage. However, once you factor in people with experience working with it, it becomes a self fueling firestorm that's hard to escape. No one ever got fired for specifying Rockwell (this isn't true, I have seen it happen, but the sentiment is true-ish).
Siemens is #2 in these regions and probably #1 worldwide. Profinet is the better protocol when compared with Ethernet/IP, but I don't have too much else to say to differentiate them.
Third place is VERY VERY hard to get a handle on because the publicly available market share numbers are all very old, some companies are small fry in PLC but huge in embedded devices in general, some companies are super regional, and some companies are huge in niche industries.
You have to also consider Automation Direct which decided to just go with affordable fixed pricing for lower tier PLC stuff. I'm sure a ton of people that would have formerly contacted a Rockwell distributor for a quote just ordered AD like they would Amazon Prime and were happy as clams that they didn't have to negotiate pricing.
Omron is a big name and people know it and it is super popular in Japan. They make most of their money from medical equipment and making hardware that places like Rockwell private labels. They're certainly in the top 10 in the US, but probably not top 4 when it comes to PLC market share.
B&R was super fast growing and appeared to be dominating Beckhoff in market share, then 3rd party adoption for EtherCAT went like wildfire (even though it is super similar to SERCOS III which was not picked up much by 3rd party devices). EtherCAT put Beckhoff on the map and I would hazard to guess that they are now growing faster than B&R based on name recognition. ABB acquired B&R recently and will likely transition their machine control stuff over to the B&R family because it is objectively better than ABB's offering. I mention B&R and Beckhoff together because they really are so much alike at the 10,000ft level as far as capability and product selection and probably market share (they're definitely rivals). At the detailed level they diverge a lot. For the record, I think B&R is the better solution and is simply saddled with a generic name with an ampersand in it.
Schneider-Electric buys up successful PLC brands and kills them because they make most of their money with electric utility type equipment and Square-D. Some of their PLCs are made by B&R, so do you count that as B&R or SE? Elau was big in the advanced motion PLC space and is probably a 10th their size now that they are SE's PacDrive brand, likely consisting almost entirely of old Elau customers that haven't switched to B&R, Beckhoff, or Keba.
Then you have Bosch-Rexroth, Lenze, Honeywell, Mitsubishi, etc. A bunch of other big old names that are struggling to keep market share against the behemoths of Rockwell and Siemens while staving off the innovative powerhouses of B&R and Beckhoff. Lenze picked up a lot of engineers from both Elau and B&R, so there is a chance we'll see something surprising come from them.
There are also niche industry stuff like Parker, Emmerson, DeltaV, etc that simply don't exist in machine control spaces that I work in, but have big following in pumping, mining, and process.
Ultimately I think AD is 3rd in US market share and I that might be a comfortable lead. B&R and Beckhoff are close at 4th and 5th (at this point, Beckhoff may be the bigger of the two). I'd put Schneider at 6th and Omron at 7th. A lot of people are going to look at the installed base and think there is no f-ing way B&R or Beckhoff are punching above X or Y big old brand. To that point, machine control is a larger market segment than the others and it is growing. Major multinational conglomerates like Coesia and Barry-Wehmiller are standardising their controls platforms on new machine designs on those two and some black box systems you don't think about use them (the majority of inspection systems have B&R panels). If I was going off the machines at PackExpo/PharmaExpo, It looked like 70% Rockwell, 5% AD, 5% Siemens, 5% B&R, and the remaining 10% were everyone else. I saw only 3 Beckhoff machines and only one PacDrive machine (and the HMI was Beckhoff). I know Siemens has more market share, but it must not be in packaging.