As a direct answer to your question I don't think it has ever been measured in terms of from ingestion to dopamine
synthesis, but once tyrosine is in the cells which convert it to dopamine would do it pretty quickly like the two conversions would both happen in less than a second.
In terms of what your going for though that 2g unless you had a severe deficiency in tyrosine wouldnt do much if anything at all since the enzymes which actively transport tyrosine into the brain are already saturated so any more tyrosine would just float around untill other enzymes in the rest of your body convert it into miscellaneous molecules. Even if it were to get into the brain tyrosine hydroxylase which is the first step converting tyrosine to dopamine is also pretty much working at full capacity so taking additional tyrosine won't cause any signifcant change in dopamine levels anyway.
If hypothetically somehow there was a severe tyrosine deficiency and it would make a difference then its the time it takes to be absorbed into the blood stream, cross the blood brain barrier and enter the dopamine neurons so the tyrosine would be being taken up into the CNS and being converted as fast as it could after around an hour. I would have no idea as to how many mg (or more likely ug) can cross the blood brain barrier at any given time given that it relies on active transport.