Since almost a month has elapsed I thought I'd provide an update.
The copper tape fix proved to be temperature-dependent. When the unit is first powered up - cold - the noise is only apparent within a couple of inches of the speaker. However when it warms up the noise increases, not quite to the original level with just the Uniden fitted part, but audible out to a metre or so.
So I decided to do one of the permanent fixes described on RR, the one involving the removal of lacquer from the front printed circuit board at the mounting screws locations. This mod is one way to solve the grounding issue that causes the hum, which the Uniden fitted part NMP200 may or may not achieve.
I had wondered about doing nothing until the warranty ran out, but decided to investigate anyway. The mod isn't for the faint-hearted or inexperienced with electronics.
Removal of the front panel and detaching the printed circuit board wasn't as straightforward as I thought it would be from RR pix and descriptions, but I got there and completed the fix without any issues.
Suffice to say that I was relieved and pleased that after reassembly 1) the unit booted up as it should and everything worked, and 2) the hum/noise is completely gone and hasn't returned, no matter what the temperature