Euphemism for the English word for human excrement (faeces in Latin)

Solution 1:

I don't know of any nouns that fit your proposed phrase, but there are a few verbal phrases that might suit your purposes.

American speakers (and possibly other English speakers) use the euphemism of relieving oneself. So, in your example, you might say:

In the morning, after he relieved himself...

Like the German phrase you mention, English also includes the euphemism to do one's business, but it usually carries an implication of immaturity toward the object of the expression:

do one's business, (usually of an animal or child) to defecate or urinate: housebreaking a puppy to do his business outdoors.

So we might say,

After my dog Sparky did his business on his favorite tree, we returned home.

Solution 2:

The phrase "morning constitutional" is ambiguously used to mean either a morning walk, or a morning "dump". So "After his morning constitutional..."

Another possibility: "After spending some time on the throne..."

Solution 3:

Unless one is accounting for his activities to his doctor in which case he'd say he had a bowel movement at 9:00 AM today, it would probably suffice to say any of the following:

He made a pit stop

He took a bathroom break

He visited the men's room

He went to the bathroom

All of these are acceptable in mixed company, though they do tend to fall under the category of TMI (too much information) - and none clearly differentiate between urinating and defecating.