Harry's birthday is July 31, 1980. His mother was killed on October 31, 1981. That makes him exactly 15 months old when his mother and father were killed.
Additionally, as pointed out by Janus Bahs Jacquet, the letter from Lily to Sirius that Harry found in Deathly Hallows explicitly states that he had his first birthday while his parents were still living:
Thank you, thank you, for Harry's birthday present! It was his favourite by far. One year old and already zooming along on a toy broomstick, he looked so pleased with himself, I'm enclosing a picture so you can see.
Babies also age at different rates, and mature at different rates. Some children are walking as early as 6-8 months, others are not walking by 18 months.
Also, as pointed out in the comments, Dudley is approximately the same age as Harry, and in the Philosopher's Stone, the following passage is used to describe Dudley:
‘You don’t mean – you can’t mean the people who live here?’ cried Professor McGonagall, jumping to her feet and pointing at number four. ‘Dumbledore – you can’t. I’ve been watching them all day. You couldn’t find two people who are less like us. And they’ve got this son – I saw him kicking his mother all the way up the street, screaming for sweets. Harry Potter come and live here!’