EMB-505 First Officer reported a loss of hydraulic pressure just prior to leveling off at top of climb. Flight crew diverted and main tires on both main gears failed during braking on landing.

Narrative

We took off from ZZZ during climb out nothing unusual occurred. We climbed up to FL410 at about FL400 we got a CAS message for HYD fault. Once leveled off we ran the QRH for HYD fault which was nothing. As soon as we completed the QRH we got 3 CAS (Crew Alert System) message- pusher fail; HYD low pressure; BRK fail. We ran the QRH for HYD low pressure first due to it being a hydraulic issue followed by pusher fail and then BRK fail. We then did the cruise checklist. I ran the numbers for the landing and called the company spoke to Scheduling and Chief Pilot. Chief Pilot and I ran through all the numbers/options and concluded ZZZ1 was the best option. I ran it passed Pilot Flying and she concurred. Once off the phone I briefed Pilot Flying and told ATC that we were diverting to ZZZ1. I asked Pilot Flying to refuse descent orders from ATC until we have everything figured out. I than went to the back and informed the passengers of the issue with the airplane and where we are going. I gave another safety briefing of wearing seatbelts during landing. When I returned upfront ATC tried to have us descend but I wasn't ready we started the decent but told them we are still running checklists at would like to stay high. I read through the checklists again to prepare for the landing. Pilot Flying and I briefed the approach. I asked her if she wanted to fly and she said yes. While I applied the emergency brake. I asked her to leave Autopilot on till about 300 feet and then disengage the Autopilot and come in lower than the glideslope to maximize the runway length. We were on the arrival and descending. I went back and briefed the passengers again and showed them how to use the emergency exits. Once we started getting vectored; we ran the QRH for HYD low pressure again to lower the gear. We used the emergency lever to lower the gear before intercepting the localizer. We intercepted localizer and continued configuring for landing the airplane (about 10 miles from runway). I briefed Pilot Flying again about coming in lower than normal to maximize the runway we have. At 100 feet Pilot Flying put the throttle to idle and removed her hand. I put my hand over the emergency brake. We touched down before the 1;000 footers. A couple seconds after we touched down; I slowly started pulling on the parking brake to slow us down. Once I felt friction; I held it. The right tire popped. I released the brake and told Pilot Flying to move to the center. We were on the right-hand side of the centerline. Once centered I raised the parking brake handle again. The left tire popped and I continued to hold the parking brake and as we got slower; I pulled on it a little more to get us to a complete stop. I got on the radio and told Tower we were stuck on the runway and are tires are popped. I shut the engines off and looked at Pilot Flying and then the passengers to make sure everybody was ok. We remained in the aircraft until airfield operations arrived. I exited the airplane and talked to airfield operations. After 40 min the bus picked us and the passengers up. I pinned the airplane up and got on the bus.Suggestions: Only thing I would have done different is [request priority handling] after talking to the passengers the first time.