but still reguardless of oil in the water (not the cause of your noisy cam tappets). if the oil pressure itself is low.... bolting new parts to the engine and flushing the cooling system is fairly pointless if your guna end up buying a s/h engine (cus of the over heating/tapping) it will most likely have a good oil cooler already on it.
check the oil pressure to assess engine condition. given the rep. these have for carbon build up blocking the oil strainer.
the oil cooler could have been seeping oil into the coolant for weeks without you noticing.... then the strainers got too blocked up causing the engine to sound tinny and tappety. the thermostat could have even stuck closed to over heat it.
you then spotted the oil in the coolant and presumed this is the cause of all your problems.
we had a 55 plate passat 1.9tdi in at work around 18 months ago, coolant had thick grey/black sludge in every upper part of the cooling system, expansion tank was full of it.
car was booked in for a service..... the guy had no idea and not reported any complaints about it running/sounding any different/over heating.
that was the oil cooler. took about 3/4 goes (bottle of cooling system degreaser each time) flushing and running it up to temp again. was a total ball ache!
the fact he had no idea this was happening to the cooling system (and it was used everyday covering high mileages)..... shows this isnt the cause of your over heating or tappets being noisy (oil floats on water so the expansion tank may look bad, top few cms, but the rest of the cooling system should be operating normally).
check the most serious suspect of your problems first. low oil pressure. cus after youve fitted the new oil cooler and running it up to flush the cooling system you could have very little oil pressure and not know it, the light only comes on at like sub 12-15 psi but you should have atleast 25psi at hot idle and up to 60psi at 2500 rpm.
ive seen blocked up oil strainers a few times and they are tappety like you describe, on checking they usually have like 20 psi at idle but struggle to even make 25 when you up the revs.