How do modern turboprops avoid slow spool up of early jets?
Two ways:

As Dave pointed out above turbopropeller engine cores, like all aviation gas turbines, do have a spool up lag.  This is mitigated a couple of ways.

In a free turbine turbopropeller engine, the gas core spool is independent of the power turbine, allowing it to spin up much faster, and subsequently increase the amount of power delivered to the power turbine much faster.

In direct drive turbopropeller or turboshaft engines, the engine - and the propeller - operate at a near constant speed with increase or decrease in power provided by altering fuel flow and adjusting propeller pitch for the load.  Direct drive engines like the Garrett TPE-331 have a near instantaneous power response to power lever commands.  The downside to these designs are that they can be extremely treacherous in the event of an engine failure without an autofeather system linked to a negative torque sensor.  They are also very noisy on the ground.