EMB-XRJ Captain reported marginal control of aircraft while performing ILS approach in turbulent weather. Captain diverted after executing a go around and later learned that the destination had a tornado warning that he was unaware of.

Narrative

At the time; we believed that our flight into ZZZ only had the threat that there would be only light rain; and possibly a few butts in the area. I called Dispatch to confirm this prior to flight. The TAF lines showed only moderately gusty conditions and a crosswind; but nothing in my and the FO's opinion would jeopardize our ability to operate into the airport that night. Dispatch even said to me before we left that the current METAR indicated just gusty VFR conditions with scattered and broken layers of butts. So we departed. ZZZ1 was the perfect alternate considering at the time and through most of the night; it would remain VFR. Even if we encountered unexpected conditions below landing minimums; we would have a great plan B. As we continued to fly further south I noticed the rain outside the aircraft had begun. It was pretty benign for the most part turbulence was intermittent light at the worst. Things began to change however as we approached the ZZZ airport. At around 30 miles north of the field we were handed to Local Approach and told to expect vectors for the LDA Runway XX. ATC informed us that there was only moderate and light precipitation and that they would vector us around it for the LDA Runway XX. Everything felt pretty normal up until we intercepted final. After settling onto final; the automation began making what I considered to be excessive bank changes to maintain localizer. This was something I had never seen before. This was also surprising considering also that while choppy; speed control was only +/-10 knots within a somewhat reasonable margin. As we continued things only got worse as the autopilot was commanding increasingly erratic banks. I decided from there to continue by hand flying the approach as the autopilot control seemed marginal to me. After a short while of me hand flying; I too realized that my own control became marginal and maintaining center on the localizer became impossible. I called for the standard go around. To alleviate a possible terrain conflict as I worked hard to maintain control; I told my PM to get vectors out of the area. He had to ask about twice before we got a response and then after that 3 more times to ask to go over to approach and get vectors back to ZZZ1. Even though we had the fuel to try another attempt I felt that those conditions were amongst some of the worse I had ever encountered in my experience. We mutually agreed that it just was not the best course of action to try again. After we got situated back in cruise flight to ZZZ1; I made my PA to the passengers and communicated with the Flight Attendant. The Flight Attendant informed me that while we were on final; her link device got a tornado warning message; which piqued my interest on the subject. After we landed I did some digging and came up with some data that supported we may have flown through a cell associated with that tornado warning. Sure enough as attached; I found that a tornado warning was issued for the area around south east of ZZZ to the area around final for the LDA Z Runway XX. Attached I also have some images of the weather during the time of the approach; as well as images detailing the approximate location of the debris ball/ approximate location of tornado. ATC or Dispatch did not seem informed of the fact that there was a cell in the vicinity that was associated with a tornado. I understand the poll rate on the radar does have a bit of latency; and dynamic situations like a tornado or severe thunderstorm are quite rapid and hard to catch immediately. But these conditions were forecast; and I feel like both Dispatch and ATC should have had an understanding that maybe some cells could be suspicious out there if they're forecasting these extreme conditions. Airports like ZZZ are unique in that the terrain greatly exaggerates the severity of certain weather conditions which I understand only further complicates the process of understanding the actual conditions. What I saw on the TAF prior to departure; looked a far cry away from what we actually encountered there on final. I did not have much other information relating to a thunderstorm watch prior to departure other than a convective sigmet for pretty much most of the east coast. I did not think to check for a tornado watch before flight; or even as we began to arrive; a tornado warning. I feel that in the future a greater focus on conditions as a whole in the atmosphere like Convective Available Potential Energy (CAPE) index; or severe weather watches; etc. and how we can inform crews before and during flight could greatly improve the margin of safety on the line.