Brian, my "new" '87 300D turbo had bad stuttering and nailing at a cold start last December (its almost a year later now and nothing bad to report in fact I really like this car a lot!) even after letting the GP's warm 30 seconds. It had lots of fuel leaks and needed lifters even with Mobil 1 it was noisy. I decided to send the head (#14) to be checked for leaks and flatness because the head gasket was leaking. At 255K miles, it had received little or no attention by PO.
After installing new lifters (following a valve job and skim cut) and new IP seals it has only a very slight nailing after only 15 secs pre-start warm up period. The noise dimishes rapidly, say within a half minute it is almost quieted down, this after 2 or 3 days of non operation and ambient around 50 - 60 degrees. The first start on a hot day the nailing is zilch. After a short distance it is completely normal. I am using a little RedLine catalyst also in the fuel (1/3 bottle at fill ups) so that may be a factor.
If your GPs are all OK, and your IP seals have been replaced, there might be something related to the lifters that is involved. That would be my hunch.
As for getting it down to a 5 sec warm up - I don't think that is possible cold. MY GP timer is about 10 seconds as I recall (I'll count next time and verify that)
As addendum, MY '85 OM617 (completely rebuilt engine) has no "misfiring" (what I would call nailing) at all even cold (what cold we have here is above freezing so that is not COLD, just not warm!). It is an almost instant start, say 5 seconds on the GP and its off and running smooth. What a gem.

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'95 E320 Wagon my favorite road car. '99 E300D wolf in sheeps body, '87 300D Sportline suspension, '79 300TD w/ 617.952 engine at 367,750 and counting!