problem with used car within 6 months

Hi, looking for some advice for my daughter.
She paid £3,000 for a Peugeot 308 at end of January. When she picked it up it was after she left work and there was only a young mechanic there to give her the keys/car, paperwork etc.....so she drove it home about 15 miles...she realised there was no heating...and so she rang them the next morning and they agreed to send a mechanic out to her to get the car, fix the heating and return it to her. All great on their part.
Also they supposedly replaced both keys as they didn't operate the central locking correctly...now, the last couple of weeks she has had to keep topping up with oil and water more frequently than you should have to, she used all her savings on this car and yesterday it just wouldn't start for ages, when it finally did, white smoke was coming from the exhaust and she could smell burning so immediately switched off the engine, now it won't start at all.
She lives out in the countryside and now has no car to get her daughter to school or herself to work. I have told her to email the garage she bought it from as they were really good with her, but is there any way they could still be liable as it is within 6 months of purchase and it wasn't a cheap runaround to her.
I live 200 miles away and she was in tears just now wondering how to get to work tomorrow x thank in advance. also, someone told her it sounds like the head gasket, does this happen quickly or over time please.