Corporate Captain reported traffic conflict on climbout from non-towered airport resulted in a NMAC and evasive action.

Narrative

Near miss shortly after take off. TCAS 'traffic' alerted; quickly followed by 'climb'; commanding almost a 20 degree nose up. By 1600 msl we were issued clear of conflict. After clear; the copilot started to talk finally. He said it was 'close'. Initially he said it was an aircraft type A; but later told me it was an aircraft type B; I think he pulled it off the internet on a replay. The issues started with a busy airport; uncontrolled with a mix of aircraft. We started our taxi as we were getting our clearance via phone. The RCO (Remote Communications Outlet) on the field wasn't working as we taxied in; so we attempted to use the phone leaving my copilot to focus on the clearance which had only a 5 minute void time. So as I heard this; I started our taxi to the runway. He finalized the clearance and loading the box etc. by the time we approached the runway. One aircraft did a touch and go; another landed; and I found Aircraft Y on TCAS which appeared to be far out to the west traveling south at a high rate of speed. I assumed he was south bound and not landing. The copilot completed take off checklist and said he was ready to go. So I continued on to the runway. As we took position; Aircraft Y which was moving fast turned final and the aircraft landing did a touch and go....slowly. Clearance void time was ticking....and the aircraft was climbing slowly. We waited until it was at the end of the airport boundary and took off. We quickly caught up to him and side stepped to the right. While focusing on the two aircraft both still climbing in the upwind (runway 13) we were paralleling them to the side; we passed the aircraft in trail of the first and almost caught up to the lead; when the TCAS alerted. Quick glance inside indicated traffic come up from the south...must have been flying up the shoreline or trying to enter the pattern on the crosswind. I never saw Aircraft Y approaching from the South. My initial response was to turn away from the aircraft and climb watching the first aircraft. Then the climb alert so I leveled and pitched up. Over in seconds. We were above all the travel; clearance was 090 MH (Magnetic Heading) and 2000 MSL; so leveled and switched to Jacksonville departure and resumed the flight. Take aways:1. don't use cell phone while taxiing. Keep both pilots alert to traffic in the area and looking around while taxiing.2. Demand a longer void time.3. Allow more spacing at uncontrolled airports for slow moving aircraft.4. Be patient5. Have the TCAS display set at lowest; mine was at 10 miles; still too much for such an event. 5 was lowest with my system.6. Despite noise abatement procedures suggesting 1000 ft per min climbs; maybe getting out of the traffic pattern sooner may mitigate issues; our thought was to be slow speed and slow climb to allow better forward visibility and slower closure rates.