At some point in this sentence the writer’s concentration wandered away, never to return:
The smallpox eradication programme was completed during the Cold War at a time when political upheavals delayed timely eradication.
Look at the logical failure at the end: was the eradication still timely after all those delays? No. Then either the upheavals prevented timely eradication or they delayed eradication, full stop.
And what on Earth is the overall structure doing? ‘It was completed at a time when things delayed it’? No, that won’t do. This hangs together much better:
The smallpox eradication programme was completed despite delays caused by some of the political upheavals of that period of the Cold War.
But it can be much shorter. In particular, while there was an official Smallpox Eradication Programme, this sentence doesn’t need to mention that as such. So:
Smallpox was eradicated despite delays caused by Cold-War political upheavals.
I thought about getting away from the passive:
Cold-War political upheavals delayed the eradication of smallpox.
But, given the context of the preceding copy, I wanted the eradication to be the opening focus of the sentence.
One thing that can get me down about my work is that I don’t have the time to give every clumsy passage the attention I’d like to. But making the effort when I can perks me up again.