my 124's nearly always started on the first crank, within 1 second of cranking, without any gas pedal. this includes a 260E and a couple HFM 3.2L's
things that could interfere with this could be ignition or fuel related.
the fuel pump should run for about 1 second when the key is turned on, to develop full fuel pressure, then shut off til the motor cranks over. a fuel pressure gauge would show you whats going on here, but that pressure should stay in the system for at least 5-10 seconds without notably dropping... on a CIS car, its around 60 PSI if I recall correctly.
the CIS system is quite robust, but if someone has mucked about with it, it may be badly out of whack. your injectors may be old and crusty (I had to replace the injectors on a low mileage 1990 260E that had stumbling issues). injectors can be tested by removing them from the engine and doing a 'pop test', where a cleaning solvent is pumped into them at oiperating pressure and you verify they don't spray anything below the minimum pressure, then have an even pattern at the operating pressures. There's a fuel pressure differential test that involves a couple pressure gauges connected to various points in the CIS fuel distributor. numerous articles on this forum about how to do all these things, I've forgotten many of the details as I retired my 124's a couple years ago.
