The temperature outside the barrel depends on the time during heating and on the location of the barrel, and is only a bit higher than the highest water temperature of the bath. Which is about 45degC on the top at the end of the heating. Lower on the barrel the temp is lower. (Mostly depending the supply temperature). Due to these temperatures I concluded insulation would not benefit the efficiency much.
The bath is constructed of concrete elements, which were originally for plants intended (on the first picture you can see the particular shape of the elements.) The bath is not insulated from the soil other than the air in the concrete elements, which I thought to be sufficient insulation, the exhaust gasses leading under the bath I saw more as an additional benefit. Thinking it would cause a small efficiency increase, the main work is done by the barrel and the pipes. I realise the insulation is not ideal however after 24 hours the water is still about 25 degC at an outside temperature of 8degC.
What I hope to learn from this tread is how much wood people need to heat their hottub for a certain temperature increase for a certain amount of water.