does john mills buy travel tickets in this movie

No - the travel tickets (honeymoon) were ordered and paid for in advance by the young couple (Hywel Bennett and Hayley Mills), but the Travel Agent did the dirty on them and absconded with the funds. Much later in the film they were sent a refund, which they used for a delayed honeymoon holiday to Blackpool - John Mills gave Hywel Bennett some cash as they departed. "Always have cash in thy pocket, lad" he said.

No, he just acts everyone else off the screen.