Although HE is July 1971 the registration date is 30th March 1972 so maybe the engine hung around without a frame for a while.
More likely it just was not sold, 1971 was a very bad year for Triumph. Your bike probably sold early in 1972 because it was discounted heavily so Triumph could try and sell the bikes they were actually building in 1972.
Indicators are on the right side (twistgrip), no power to test because of the short but switch goes both up and down on right bar and only middle or up on the left bar. That makes it '71?
Handlebar switches with the headlamp dipswitch, cylinder head exhaust spigots, individual rocker access caps were all 71-72 changes.
With the terminals off the coils measure 3.8ohms DS and 4.1 ohms TS so you say they are 12v?
Yes.
wiring
I can’t believe the plastic loom wrap is fifty years old … I don’t know when they stopped using fabric wrap
My feeling too the loom is not original, but possibly an original Lucas spare fitted with the Rita, possibly in the 1980's? Lucas was making spares but there was not the current fixation with "original", so Lucas would have used contemporary production techniques - e.g. plastic tape wrapping.
I wasn’t aware of the parallel/series distinction
which way are mine wired?
In parallel.
Any electrical circuit is between -ve and +ve and must include an electrical resistance or it is a short circuit - lots of Amps, melted wiring, fire, etc.
'Connection in parallel' mean each component has its own connections to battery -ve and to battery +ve; ignition components are 'in parallel' with light bulbs, are in parallel with the horn, etc.
'Connection in series' means more than one component but the whole series has just one connection to battery -ve and one to battery +ve. Coils 'connected in series' have the e.i. switched wire connected to just one coil's -ve terminal, that coil's +ve terminal is connected to another coil's -ve terminal, that second coil's +ve terminal could be connected to a third coil's -ve terminal, and so on, only the +ve terminal of the last coil in the series is actually connected to battery +ve.
Multi cylinder engine with a separate ignition coil for each cylinder switched by points, the coils are connected 'in parallel' with each other. Your bike has one White/Yellow wire from the handlebar engine kill button, it was originally connected to one coil's -ve terminal, the WY wire spade terminal had a second WY wire crimped into it, the other end of that second wire was connected to the other coil's -ve terminal; each coil's +ve terminal had a separate wire to one set of points, one side of each set of points was not insulated from the points plate, the points plate was in electrical contact with the engine, on something like a cylinder head bolt, a Red wire is connected, the loom Red wires are interconnected and to battery +ve. So each ignition coil had its own connection to battery -ve through the kill button and ignition switch, had its own connection to battery +ve through its points, the engine, loom Red wires.
Currently, your bike's Rita connections mimic the above, except any electronic ignition is always connected on the -ve side of any ignition coil. Your bike's White/Yellow wire from the handlebar engine kill button is connected to the Rita Black -ve wire, the Rita's White/Black switched wire is connected to both coils' -ve terminals, the Rita's White/Yellow +ve wire and both coils' +ve terminals are (should be) connected to battery +ve.
Points switch only the connected coil on and off individually, so multiple coils are not all on and charging at the same time. The downside of the current Rita connections compared to coils switched by points is Rita switches all connected coils on and off at the same time, all off when any cylinder requires a spark ("wasted spark"), all on and charging at the same time; this is why your bike's Rita connections cause such high Amps draw.
Because Rita switches all connected coils on and off at the same time, it is better to connect multiple coils in series. Components connected in series, their individual resistances are cumulative, you know a 12V coil's primary resistance is ~4 Ohms, a 6V coil's primary resistance is approximately half that, so if you connect two 6V coils in series, they will have the same total primary resistance as one 12V coil, so the Amps drawn by two 6V coils will be about the same as one 12V coil.
The flat box looks relatively new in both design and condition. Would that be the rectifier?
"flat box" is:-
... if so, this is where you could become very confused ... it is
a rectifier but it is not
the rectifier on your bike ...
the rectifier is the circular multiplate thing to the right of the "flat box" ...:-
. if you look at the wiring diagrams you have labelled "1971 T120" and "1972 T120", "RECTIFIER" is depicted as four arrowheads with one point against two lines perpendicular to one another;
. each "arrowhead with one point against two lines perpendicular to one another" is the standard symbol for a diode;
. alternators generate Alternating Current, which primarily cannot charge a battery or work electronics, four diodes connected as depicted convert Alternating Current to Direct Current, which can both charge a battery and work electronics;
. the Rita wiring diagram shows two diodes, in the White/Black wire between the amp and each coil's -ve terminal;
. your photo. shows the White/Black wire from the Rita connected to the "flat box" terminal that would be an AC terminal if it was being used as rectifier, the other two White/Black wires are to the coil -ve terminals, the two diodes inside the "flat box" between the terminals are the two diodes represented in the Rita wiring diagram.
I suppose the terminals tucked into the fender cross support are left over from the old points system but as they disappear into the taped loom I can’t trace where they go
British twin cylinder bikes, two coils switched by points, Black/White wires connect the drive side coil +ve terminal, condenser and points; Black/Yellow wires connected the timing side coil +ve terminal, condenser and points.
On OIF, the condensers were mounted between the coils; your bike's drive side condenser mounting hole is visible in your photos, timing side condenser mounting hole now mounts the bike's rectifier (circular multiplate thing). Black/White and Black/Yellow wires consisted of a short wire from the coil +ve terminal to the condenser male spade terminal, female spade terminal had a second wire crimped into it that went to the points.
With Rita (most e.i.), the Black/White and Black/Yellow points wires are repurposed to connect the electronics box to the trigger unit in the original points compartment in the engine timing cover.
I noticed that mine is a Rita AB11 47270B whereas all photos I saw on Gurgle were 47270A, were these for Nortons and the 'B's for the Triumphs with their different 38 deg advance?
Nortons were never fitted with Rita as original equipment.
Lucas A, B, etc. part number suffixes was their system to show revisions - A was the suffix of the first production version, B was the first revision, etc.
Most (all?) Rita amps have one advance curve, it is modified for different bikes by the trigger unit fitted.
I have added yet more photos at different angles showing the flat box between the coils and, yes, the round object is indeed the key switch.

Thanks.
I put the dead fuse back in the holder to keep the sprung terminals apart …… without the dead fuse the contact ends touched in the holder and started to melt the white and brown/blue wire either side,
Is the ignition switch in the off position? This is not when the key is turned fully anticlockwise, then the pilot and rear lights are on, known as "parking lights".
Everything off is one click clockwise from fully anticlockwise. If the fuse blows with the key in this position, the short can only be between battery -ve and the ignition switch. Disconnect the Brown/Blue wire from ignition switch terminal 1 to eliminate the possibility of a faulty switch. Then the most likely candidates for a short with the ignition switch off are the rectifier and Zener diode. Disconnect the Brown/Blue wire from both, fit new fuse, reconnect one at a time, with a short but noticeable pause between each reconnection. If the fuse does not blow when rectifier and Zener diode are reconnected, reconnect Brown/Blue to ignition switch terminal 1.
way to find the short in the loom?
After the previous two paragraphs, still easy, if convoluted the first time.
Easy because you just connect a light bulb in place of the fuse.
Convoluted the first time because:-
. you need to acquire another fuse holder the same as the existing one;
. add one part of the new fuse holder to each end of the bulb holder's wires;
. then the bulb holder can be connected between the two parts of the existing fuse holder.
Once you can connect a bulb in place of the fuse, start by disconnecting all the wires from the ignition switch, reconnect one at a time in the following order; when the 'fuse bulb' lights, you have located the short:-
. turn the ignition key fully anti-clockwise, reconnect the Brown/Green wire(s) to ignition switch terminal 3;
. turn the ignition key two clicks clockwise (one click clockwise being all off), reconnect the White wire(s) to ignition switch terminal 2.