Engine rattle at start up

look for any metal or plastic cover/protector for casing of parts/ moving parts or the radiator fan casing rattles are majorly caused by poor bolting of parts or cracked parts. . best to do this with a stick and to dip in the parts with caution and see if rattle disappears. hope this helps..

Does the car make this sound when the car has just been turned off and engine is still hot? Lightly push the car back and forth while it's in park. Does it still make that sound?

not the water pump as the noise will remain constant hot or cold
you leave very important details like make . model , year and engine type petrol or diesel so answering becomes a hit and miss affair
if the engine has a timing chain it indicates that the hydraulic adjuster is loose until oil pressure is built up which equates to a badly worn chain
next if an incorrect oil filter was used , the oil will drain back over night and , chains, rockers, hydraulic lifters will all rattle until pressure is built up
incorrect oil viscosity will drain down quickly and will take time to build up pressure as well
you may have to find an accredited service shop and leave it there over night so that the diagnosis can be made in the same conditions for them as for you and then get a quote

READ THIS (happy ending so far...):
audi a4 2L turbo quattro, manual transmission with 98,000 KM. Perfect conditions, no oil leaks whatsoever.

Left the car off for two weeks in very cold weather (-10C down to -25C). When I restarted, it started immediately but with heavy rattling noise due to camshaft adjuster not getting enough oil pressure. After 3 minutes of warming up (outside temp.: -12C) at very low rpm (below 2,000rpm), I start moving the car. Still hearing heavy rattle noise. As soon as I pass 2,000 rpm, I get a scary red warning, low oil pressure. Turn off the engine immediately. In 1 minute, I re-start the car and the engine lights check ok, so I drive in first gear super slow (1,200rpm) hearing the same rattling noise. After another minute, finally the noise all the sudden disappears and the car starts running just fine.

Called Audi dealer and they told me that in extreme weather conditions, I should warm the engine for 10-15 minutes if necessary before start moving.
Take away: luckily, I didn't screw up the engine since I was using it at extremely low rpm (easy to do with clutch and manual transmission) until the oil reached full pressure. I am pretty sure now that the problem was leaving the car off for such a long time in freezing weather. All the oil flows back to the bottom, and when I restarted it, the cold oil with high viscosity took a long time to reach all the parts of the engine, specially the camshaft adjusters that are actuated by using oil pressure. Next day, the car started at -9C perfectly without making any noise (as usual). Basically, the longer you leave your audi in cold weather, the longer you might hear the rattling noise until the oil reach full pressure inside the engine. My car took 6 minutes to fix it after couple of weeks in cold weather. This makes sense to me; in fact, in the past I only heard the rattling noise after leaving the car off for a couple of days in the winter.
So far, this has had an happy ending....

fine metal shaving means major engine damage, time for a tear down , start with the oil pan, find metal in there which you will, the damage is extensive, time for a rebuilt

Check all braces under your car. Not just visually, put hands on each and every brace. Sounds like one might be loose or broken. Hope it's as easy a fix as mine was.

make sure its not a cracked fly wheel

4.0l's are notorious for valve rattle on cold start-up. There is really nothing you can do to make it go away short of completely replacing the valve train in the top end, and that isn't a permanent fix. It's is an top end oiling issue from the factory. Just don't race your engine on cold start-up until the rattle stops.

The rattle in the transmission sounds to me to be more of a loose dust shield/heat shield around the transmission or exhaust than a mechanical issue.

possibly the cat itself broken up and rattling in side the silencer especially if more noticeable when engine warmed up.

It has been suggested to me the problem could be a belt tensioner gone bad. Sound right to anybody?