Newels should be long enough to extend down to the joists and be through bolted to the joists or solid bridging.
I use x4 lumber and at last ½ inch fasteners in bored holes.
The holes should be undersized enough (1/16 to 1/32 below the fastener) to require driving the bolts.
Only unthreaded shank should be in the bearing area.
2 inch washers under the bolt head and the nut are a good idea to avoid crushing the wood.
Steel angle iron can be used to fasten bridging between joists if required.

An 80 pound load 4 feet from the first fastener produces an almost 500 ft-lb torque.
The wood cells structure can be crushed and then the newel is loose. The damage accelerates from there.
How many times have you seen a 120 pound child swing around the newel at the bottom?

Does anyone know how to re-thighten an existing 'round' newel that has come loose. Tailing as a result moves about 1 to 2 to the right+left. I found stabilizer brackets for square base newels but nothing for a round base newel. Any advise would be greatfully appreciated.

"Does anyone know how to re-thighten an existing 'round' newel that has come loose. "

Every round newel I have ever seen had a square bottom section for attachment.

It may be attached to the stair carriage under the last step if it is not correctly attached to the joists below.