There seem to be many articles written about Atmos clock repair, a quick google reveals several pages but none seem to face the actual task of re-filling the bellows.

some of the many atmos hyperlinks referenced

www.abbeyclock.com

kitztech.com/bellows.htm

atmosman.com/bellows

clocks-uk.com/Atmos.html

David Moline & John Wagner 'Operating Principles, Common Questions and Performance Data for an Atmospheric Driven Atmos Clock

The latter is a very interesting technical article which, if it can't be sourced at the above hyperlink, is well worth searching for to download as a .pdf from Clemson University Tiger Prints mechanical engineering by David Moline & John Wagner entitled ‘Operating Principles, Common Questions and Performance Data for an Atmospheric Driven Atmos Clock’

Of the many Atmos clocks I have been asked to repair, I’ve found wind /bellows related issues to be a rarity - I would be interested to hear from other repairers whether this matches their experience.
My last repair proved an interesting diversion. I had serviced the clock back in 2002 & thought it an unusually short interval of return. Because l couldn't find a source for replacement/repair of the bellows, I researched the problem. The bellows appeared undamaged, they just no longer responded to temperature change - with this in mind, I decided to try to refill them myself.
On removing the bellows assembly, I was interested to note that one of my google searches revealed a suggestion that the date of bellows filling/manufacture is written on the bellows facing. Using an 8mm ring spanner to remove the 2 nuts on the model 526 revealed a date of 16th March 1962 so it had lasted for over 50years. On removal, the wind capsule was fully collapsed. Manually winding the clock (by extending & collapsing the wind arm) provided power to the clock which worked without a problem & proved that the failure was indeed due to the bellows.
As a safety measure, the bellows/cover was chilled before rotating the bayonet cover & releasing the bellows themselves. Placing the bellows in water & compressing them failed to reveal any stream of bubbles that would indicate a leak.

I’m no chemist but my mind tended toward assuming air present when the Ethyl Chloride was added , slowly reacts & alters the chemical composition of the filling until it fails to react to atmospheric/temperature change, a colleague was of the opinion that the small molecule size of Ethyl Chloride makes the metal semi porous allowing a gradual evacuation of the bellows - subsequent testing proved both these to be incorrect - the bellows had breached at the junction seam. At 24 C the pressure with 5ml of Ethyl Chloride was sufficient for this to be seen when submerged. The bubble stream was almost impossible to see but present, the pressure loss sufficient to fail the bellows.
Because soft solder has been used to attach fill nozzles, base & facing, I felt that this medium gave the best chance of continued reliability, this was the preferred option.
Next problem, the bellows skin is a white metal, standard flux/ solder showed no enthusiasm for letting the solder flow. Testing the surface showed it inert to gold testing fluids & non magnetic so I was fairly confident it was stainless steel. The necessary flux is acid based & had to be couriered but worked like magic, it took several re-fills to establish the minimum safe cover with several tests failing & necessitating the whole process to be repeated but the learning curve helped me polish the necessary skills.
Because the re-filling is completed in house, I can personally guarantee the repair & complete it expeditiously.
Whilst several contributors note that not every bellows can successfully be refilled, I am confident that in most cases we can successfully complete this service & I’m happy to answer enquiries regarding this service whether from a member of the public or a trade repairer