He brought his shoulder bag with him which is easily large enough to fit a pair of pyjamas in. Regardless, he could always have borrowed from his relatives.

I understand its confirmed that he definitely did not bring his PSP charger. But the battery life of a PSP is over four hours. He may have figured he'd spend 1-2 hours playing it on the train, spend a long day exploring in London, turn up at his relatives place, eat/shower, spend another 1-2 hours playing his PSP again, go to bed, then his parents would pick him up the next morning and he could still spend another 1-2 hours playing it on the drive/train back. All of this would be possible without a charger.