A320
Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Flight Crew Type 390
Communication Breakdown; Workload
ATC Issue All Types; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Fuel Issue; Inflight Event / Encounter Weather / Turbulence
Flight Crew Returned To Clearance; Flight Crew Executed Go Around / Missed Approach
Human Factors; Procedure; Weather
Flying ORD with ZZZ as an alternate. We were outside the final approach fix to 27R beginning the configuration process when we noticed the following [traffic] go around. The weather was at least 20 north of the field so I didn't think anything of it. We were fully configured and got a reactive wind shear warning and went around (disclaimer the Captain was convinced it was predictive WS however I don't recall the aircraft saying WS ahead). I followed the [applicable] guidance then cleaned up afterwards. ATC provided zero help; initially they asked if we needed help but a few seconds later changed their minds on heading and altitude at least 5 times (maybe more) increasing our workload significantly. At one point; we were climbing to 5;000 and they asked for a level off at 4 when we were passing through 3;700 (and obviously overshot it) and I disconnected the AP and lowered the nose. With the AP back on we decided to divert to ZZZ1; our planned ALT was on the other side of the weather and both of us didn't want to fly that direction. The CA (Captain) gave me the plane and spoke to dispatch and let them know of our intentions; get weather and NOTAMS and request landing data. We landed uneventfully on XXR with zero landing data (we [alerted] min fuel; had no time to wait around for dispatch to send it to the aircraft). We landed in ZZZ1 with 4.3 and taxied to a remote pad where we were refueled.
A320 First Officer reported an altitude overshoot due to ATC issuing several heading and altitude changes during heavy workload missed approach.