Mapping London postcodes onto London boroughs - doable with Google Fusion tables?

Everything I want to do below I can already do manually on my computer. I'm looking for a "yes it's here Jo" off-the-peg solution, but not really looking to learn how to do something new at this stage - maybe later :)

As I'm only likely to do this process once or twice a year I'm caught in that "is it worth expending ages learning something now to make this simpler?" Possibly not. 

If it looks too fiddly then I may well have to carry on with my manual process. Here's a post I wrote previously (with pics and worked examples) on how I used Google Fusion / tables to map UK cinemas, for the Sherlock special.

Is it easy to map a list of postcodes to London boroughs?
If it's fiddly and difficult then that's no good because I already have a low-res working method. It's just inefficient and non-shareable. I have an unshakeable belief that lots of people have probably wanted to do this and someone must have already come up with a solution for the non- or only mildly-technical user.

Background
I have a list of 100 London schools' postcodes (it could be 10, 50 or 2,000 as this needs to be scaleable for others to use) and I would like to display them on a map that says something like: 4 are in Harrow and 16 are in Greenwich, with a magnitude indicator of some kind. I might get it to do "if fewer than 5 show as blue", "if more than 5 show as green", "if 5-10 show as orange" etc). In fact I can do this fine on a PC with a bitmap file of a black and white line drawing of boroughs (using the fill tool on free Paint.exe, no idea what free tool works on a Mac).


Currently I can, with ease, use Google Fusion tables to convert a single column of postcodes to a geocoded map. It looks like a map with one red (the default, can be changed) dot per postcode.

But even at sensible magnification they eye does not naturally see the borough limits and I'd like to (a) add them (as an overlay?) to begin with - at a minimum it should be possible to see the borough boundaries (at least people can count the dots)
(b) see if I can get Google Fusion to combine SE1 1AA and SE1 2ZZ into "two in Borough X" - my workaround is to add a column and write in the borough, re-order and count them and then have two columns with the borough name in one and the number of times it appeared in the other ("Lewisham, 2" / "Barking and Dagenham, 1")
(c) make the raw file available (without my postcodes which are my data) to anyone else who finds themselves with a list of London postcodes and a desire to map them in this way.

Is there a pret-a-Fusion look-up table or a ready-made-London-borough-map that will work with a Google Fusion table of postcodes that I can ust add? I did look but couldn't find it, nor could I understand what I could find ;) I don't really know what KML or SHP files are.

Further reading

• Is it possible to add a Polygon layer to a google map v3 without using fusion tables (2013) - this is too technical for me and seems to involve using APIs and KMLs and I have no real familiarity with either 
• Map of London boroughs - but I've no idea how to add my data to it
• London Boroughs - Boroughs KML - looks interesting but no idea what to do with it