To make a long story short, I am worrying about whether wood mulch
(Cedar Nuggets from Home Depot to be precise) is safe for cats to run
on or jump onto?

Good question.

If a tree had a great deal of injections and systemics applid there is a
good chance that thge chemacls may be stored in the wood. Trees load, store
and then use.

What I use at clients properties is composted tree trimmings. This consist
of wood chips, leaves and needles. I let them compost for at least one
year. NOT FRESH CHIPS. I do not find myself recommending to clients to buy
bags of mulch. What ever you use here are some helpful tips in proper
mulching.

Proper Mulching - http://home.ccil.org/~treeman/sub3.html
and
http://www.treedictionary.com/DICT2003/M/ Look up "Mulch"

Here are two links addressing some of the chemistry of mulch.

Troubles in the Rhizosphere
http://www.treedictionary.com/DICT2003/shigo/RHIZO.html

A Touch of Chemistry
http://www.treedictionary.com/DICT2003/shigo/CHEM.html

Sincerely,
John A. Keslick, Jr.
Consulting Arborist
http://home.ccil.org/~treeman
and www.treedictionary.com
Beware of so-called tree experts who do not understand tree biology.
Storms, fires, floods, earthquakes, and volcanic eruptions keep reminding us
that we are not the boss.