It sounds like you’ve done a lot of detailed investigation, and you're spot on about the complexities of setting the timing via the cam sprockets rather than using the crankshaft markers. Let me try to untangle some of this:

### 1. **Timing Based on Cam Sprockets**:
   - You’re correct that since the camshaft rotates at half the speed of the crankshaft, **10 degrees on the cam sprockets** translates to **20 degrees on the crankshaft**. If the correct timing spec is **6 degrees BTDC (crank)**, this would indeed require only **3 degrees on the cam sprockets**.

### 2. **Understanding the Discrepancy**:
   - The **RS 1600 spec** being **12 degrees BTDC at 1500 RPM** is for that particular engine's design, but **Caterham's 6 degrees for the BDR** is more conservative, likely tuned for road use rather than performance.
   - Setting **10 degrees on the cam sprockets** (equating to 20 degrees on the crank) is likely **too advanced** for the engine based on the BDR's intended setting of 6 degrees crank timing.

### 3. **Crankshaft Pulley Marker Issue**:
   - If you're unable to reference the crank pulley markers due to positioning or potential refitting errors, it makes sense that you'd have to rely on the cam sprockets. Since the crank pulley might be inaccurately positioned, it could lead to confusion about actual crankshaft timing. 

   - Refitting the pulley randomly, as you mentioned, could have caused the timing marks to be misaligned, making it nearly impossible to use the traditional crankshaft reference.

### 4. **HC as an Indicator of Timing**:
   - As you noted, **HC levels** (hydrocarbon emissions) are useful because they indicate how much unburnt fuel is leaving the combustion chamber. 
   - If you're able to reduce HC levels by slightly retarding the timing (bringing it closer to 6 degrees BTDC on the crank, or 3 degrees on the cam), that suggests more complete combustion and proper timing.
   - If HC levels rise as you advance the timing, it could be a sign that the ignition is too advanced, resulting in incomplete combustion or pre-ignition.

### 5. **Next Steps**:
   - **Retard the timing** closer to **3 degrees on the cam sprockets** (equivalent to 6 degrees crank), as per the BDR spec. This should get the timing closer to what's optimal for your engine, and then observe both the engine's behavior and the HC readings.
   
   - **Test and adjust gradually**, observing how the engine responds, particularly under load. A slight improvement in performance coupled with reduced HC readings could confirm you’re heading in the right direction.

### Mental Rescue Summary:
Yes, you’re working in an **unusual situation**, setting timing based on the camshaft sprockets rather than the crank. The key is to **halve the crankshaft timing spec** when working on the cam sprockets. Since you're at 10 degrees on the cam, reducing it to 3 degrees on the cam (which equates to 6 degrees on the crank) should bring you closer to spec.

By adjusting and observing HC levels and engine response, you’ll be able to find the sweet spot for your engine.

Let me know how it goes.

ok so I reduced the ignition timing to as close as I can estimate on the cam sprockets to 3 degrees (which is 1/3 of a "tooth" ; (where a tooth is one tooth plus its next gap)), i.e. 6 on the crank.

To my pleasant surprise we now get almost the best emissions figures, the HC is higher than best I have had before, but CO2 and CO are better and lamda is nearly 0.9 . So i have arranged the gas test for tomorrow. That said i have the mixture screws turned almost all the way in. I started at zero and unscrewed instead of starting at 2.5 out and going in. I think tick-over needs a tweak faster.

The middle two are a bit rich, but with these figures I will wait for another day.

and the strobe RPM reading won't photo at all. 

The timing marks are hard to see/video (this is a screenshot from the video)  in bright sunlight

I was just going to say, can you not add timing marks to the cam sprockets - tipex (other correction fluids available) is very good at giving you a white line to strobe.