in the good old days cars had a manual choke you pulled and pushed blocking air as needed until warm
most modern car ecu's will instead overfuel on cold startup effectively choking the engine of air keeping it running 'normal' until warmed up.
to tell what fuel and air is needed and when, the ecu will read off various sensors inlet temp/mass air flow/manifold pressure/lambda amongst others all going like crazy but when cold it relies on a water temp sensor (inj temp) to tell it what temperature the engine is at and with clever thinking it works out what fuel/air etc it needs
if the inj temp sensor isn't reading right, and they are known to be faulty a lot, the ecu will act one of two different ways
it will run fine cold and overfuel when warm - temp sensor is saying its cold
it will run crap when cold but be fine when warm - temp sensor is saying it's warm
your AEE is possibly monomotronic (single point injection) so will have, i would imagine, a round 4 pin sensor in the top of the housing the radiator hose connects to - usually clipped in and black with a yellow ring
2 pins run the gauge (keep eye on that) other 2 send ecu a signal
it's relatively cheap and for a cold start
problem like yours i would suggest changing or at least testing that first
once you get rid of any possible air leak problems first mind or you'll be on a hiding to nowhere