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Lily had a panic attack of happy when Dumbledore showed up, her emotions all over the place. She would've tried to pick sans left arm James up and carry him toward Dumbles and the portkeys, but would've needed help (Sirius?).
She's mostly just minor cuts, scratches, bruises and etc minus her serious burns, which someone would've had to have thrown paste/salve on before her contractions got too bad.
Most of her time in labour would've been spent crying and worrying about James. After giving birth to Harry someone probably tended to her burns and wrapped her up, then gave her a dreamless sleep potion and put her in a makeshift bed downstairs (since the house has been locked up for nine years and probably needs a good cleaning).