I'm not saying for certain but your chain tensioner spring could be bad. When the oil pressure builds up do you notice the noise simultaneously going away? If so, you need to replace the spring in the tensioner if you're up to it, or the tensioner all together. I had this same problem. When your car is warm it will build oil pressure quickly enough to not make this happen.

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'82 240D 224K miles manual transmission
mods: wooden 4by4 bumper, EGR delete and older EX manifold without EGR port, glass pack muffler (cheapest replacement muffler), rebuilt bosch injectors with Monark nozzles

working on: aux electric fuel pump, coolant/fuel heat exchanger/filter head, afterglow, low oil pressure buzzer/LED