Thanks for all the input from you all.Some more info that might help unravel things and especially in light of Robin's post.

Robin (and I love the attitude of anything can be made to work!!:) -
Thank you I will indeed do that (in fact have). Canberra is a long way. Though a friend of mine is off to New Zealand soon and could perhaps throw it out the window on the way (not sure if you can open the windows though)

1 It was four strings (and may be again when I change strings) but currently three. Tuned D A D. String gauges 0.22 0.12 0.12
2 It has 15 frets - 8 up to and including the octave and then another 7
3 No zero fret

I don't know where it's from. It has Appalachian on the headstock. The headstock is weird in that the four tuners are in a row and that doesn't help. There is a picture of it here. It wouldn't surprise me if it's made out of a packing case :). But it is what it is and I will get a tuneful sound of it...

The intonation at the octave seems a little out. It's sharp compared to the harmonic. To give an idea how much. If I tune the D string to D 440 and play the harmonic so it is in tune; then when I play the fretted octave it's about 25cents sharp - or I would need to change it to 446Hz to make the D in tune. If that makes sense. And same for A string. If I play the low open D and play the third fret on the A string (which I reckon should be a D) then again the fretted A string is sharp again about 20+ cents; again to make it in tune I would need to change the tuner to 446. The tuner is not fantastic but gives an indication

The action seems high to me - at the octave about 5mm and at the first fret 3mm

At some point in the past I think that I might have turned the nut and bridge the other way round and so may have slightly shortened the strings. But I think I was already having problems - especially playing chords it was/is horrible.

I have put a video up on youtube which may illustrate where I am currently and my current (very heath robinson) exploration into ways of trying to deal with the problem to get it into a playable state
My dulcimer and HR solution

Intention next is to restring it and I may flip the nut and bridge back which I think will slightly increase the string length that may reduce the sharpness a bit. Can't be worse (or can it?). I'll report back. The string sets I have are all the same gauge so it will be as before. Having said that it is what Martin say they are for! And for tuning to D A D. Silverplated steel and Nickel wound

By the way what I have done with the green toothbrush bridge seems diametrically opposite to what I should be doing! As I'm in effect shortening the middle string. Puzzled

Here are the fret distances for sake of completeness (not exact but close to about middle of fret)

Nut to bridge - c671mm

Frets

          Calculated        Bridge turn        Actual dulcimer
                          length c675mm        
   1        73.2                  73.6                73.5
   2        138.4                139.2                139
   3        168.3                169.3                169-170
   4        223.2                224.5                224
   5        272                  273.6                272
   6        294.4                296.1                295
   7        315.5                317.4                316
   8        335                  337.5                337
  
   9        372                  374.3                373.5
  10        404                  407.1                406.5
  11        419.6                422.1                422
  12        447                  449.7                449
  13        471.5                474.3                472        
  14        482.7                485.5                484        
  15        503.25               506                505
  

If I turned the bridge the other way I think the distance nut to bridge would be 675mm and I popped some figures in but I think I'm guessing and floundering around in the dark a bit!