Nick Barrit, now 66, was 24 in March 1979 when he bumped into Nilsen on the platform at Waterloo train station.
With 36p in his pocket, Mr Barrit had missed the last train home to Christchurch, Dorset, by seconds.
Nilsen approached him and offered to take him for dinner – claiming he too was in the same predicament. Mr Barrit, a self-employed gardener, accepted Nilsen’s offer and the pair walked to the Strand Cafe in London’s West End.
After tucking into beef burgers, Mr Barrit claims Nilsen invited him back to spend the night on his sofa and became “aggressive” when he rejected the offer.
Mr Barrit only realised who Nilsen was decades later – in 2006 – when he watched a documentary about the notorious serial killer and necrophile. He said the release of the ITV drama Des this week brought all the memories flooding back, and he realised how lucky he was to escape.
He said: “I was in a bit of a pickle and he sort of came out of nowhere. He told me he’d missed a train too – although he later admitted he hadn’t – and said: ‘I’ll buy you supper’.
“I had very little other options so I decided to go with him. I went along and we walked to a cafe for beef burgers, chips, peas and carrots.
“I remember him speaking in a soft Scottish accent. He kept staring at me and didn’t say much. He seemed a bit agitated. He kept getting a cigarette out to light and then putting it back – hesitating. After dinner he invited me back to his flat in Muswell Hill to stay on his sofa. He was insistent, saying he’d pay for a taxi back to the flat and then would pay for me to get a cab to the station in the morning – but I was worried I wouldn’t make my early train on time.
“As soon as I went to go he got quite stroppy about it – bordering aggressive. But I thanked him, shook his hand and started walking back to Waterloo. Now I dread to think what might have happened if I’d gone with him.”
Mr Barrit had driven from his home in Dorset to Derby to watch an Everton football match, but it was called off when the floodlights failed twice in 15 minutes.
All the streetlights were affected too, so Mr Barrit could not find his car. He phoned his father and asked him to pay for a train ticket home.
“I keep getting flashbacks,” he added. “It makes me feel terrible now, but obviously at the time I never suspected a thing.”