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On a global scale, all time, I suspect the record will be weeks, not days.
On a slightly different topic, in 1939, on the outbreak of war, BBC television was summarily shut down, mid programme. On it's post war resumption in 1945, they started with the same programme. So anyone watching in 1939 had to wait almost six years for the end.
In 2003 I was on a train from Aberdeen to Birmingham. The train in front of us broke down, then a train behind us broke down. I got to New Street about midnight, which was 8 hours late. I've no idea what time it got to Plymouth or Penzance.
Not in the UK, but in the mid 1980s I was on the California Zephyr from Chicago to Denver which had a series of delays, including the loco catching fire and being replaced by a freight engine.
We trundled across the plains and got to Denver well over 6 hours down.
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