The startup of the BE-3 is slower because it uses the Tap-Off cycle. Hot gas from the main combustion chamber is tapped off to run the twin turbines that power the fuel (LH2) and oxidizer (LOX) turbopumps. In the BE-3 engine diagram you can see that each of the turbopumps has its own turbine and exhaust.
(source)
Startup sequence: Fuel and Oxidizer at tank pressure are fed directly into the combustion chamber and ignited. This results in low pressure combustion in the chamber. Hot gas expands and flows out of the nozzle, but upstream of the nozzle throat a tap-off opening allows some of the hot gas to flow to the manifold which feeds the gas turbines for fuel and oxidizer and starts them spinning up.
When the turbines spin up, they begin to turn the fuel and oxidizer turbopumps, which feed Fuel and oxidizer at higher pressure into the combustion chamber. Ignition already took place, so the higher pressure fuel and oxidizer continue combustion and increase the pressure in the chamber. When propellant begins flowing from the turbopumps into the chamber the direct feeds from the propellant tanks close.
The higher chamber pressure in turn flows more hot gas at higher pressure to the manifold that feeds the turbines, increasing their speed and power, which in turn increases the speed and pressure of the turbopumps. This creates a positive feedback loop as increased pressure from the turbopumps increases chamber pressure. A control valve in the tap-off plumbing restricts the maximum flow of chamber gas to the turbines, allowing them to spin up to their full speed but not beyond.
I could not find documentation on the exact sequence of engine throttling, but it is clear on the diagram that the main fuel valve is downstream from the turbopump. On the opposite side of the engine there are two more valves, one the main oxidizer valve and the other is a valve on the chamber tap-off to the turbine feed manifold. Apparently the engine can throttled using a combination of all three valves.
So in conclusion, it is the tap-off engine cycle and the process of initial propellant feed to start the combustion process at low pressure to spin up the turbines, then gradually increase propellant pressure to bring the turbines up to their full speed and finally open the propellant feed valves to bring the engine up to full thrust that is observed as the delay from initial ignition.
I'm fairly certain this is over-simplified, but I think it's fairly accurate in broad terms.
Note the new BE-3U upper stage engine for New Glenn uses a single turbine on a single shaft with both turbopumps, and uses the Expander Cycle instead of the tap-off cycle. To learn about the Expander Cycle look up the RL-10, an engine that's been around for 60 years that uses that cycle. It's only about 1/7th of BE-3U's thrust, but the cycle is the same.
