Can Naegleria fowleri enter through wounds into the bloodstream?
No, Naegleria fowleri is a free-living excavate form of protist that lives in warm fresh water. Fowleri finds its way into the brain by eating through the olfactory neurons in the nose where it multiplies itself greatly by feeding on nerve tissue. 

Once it penetrates the nervous tissue, fowleri's feeding  results in significant necrosis of and hemorrhaging in the olfactory bulbs. The protist then climbs along nerve fibers through the floor of the cranium via the cribriform plate and into the brain.

Since the protist travels by feeding on nervous tissue, fowleri could not enter the brain via bloodstream.

Just some more facts about Naegleria:


Death usually occurs within 14 days of exposure due to destruction the autonomic nerve cells of the medulla oblongata.
Luckily for everyone this disease is rare. There were only 300 cases as of 2008