If he's going to drive her once on the weekend, I would be more inclined to pick her up from work on a Sunday evening (especially depending on how busy the streets were and how dark it was when she finished work.)
I would also expect that the OP learn how to drive. It would be taxing being the only person in a household who could drive, especially if then your whole weekend was filled with "need to do groceries, need lifts to this" etc.
But if the DH is so tired working Monday-Friday, I imagine the DD would be tired too (assuming it's 5 days a week work) if she has to walk home late in the evening after work.
I don't have a problem with teaching your children to be independent and to look after themselves, etc, and yes, if she's got this job then it's her responsibility to get herself there and back. But one night a week, on a Sunday, when the buses don't run late...is it really so much to ask that a father pick his daughter up from work? And push his Sunday night drink back half an hour? My concern would be that this is teaching DD that you only have to look out for yourself, and that family members shouldn't bother doing things for each other unless there's a trade off (OP getting the paper Saturday mornings.)
Also, I don't know how it works in the UK, but could DH use these once weekly trips as driving lessons for DD?
That being said, DD is an adult. Maybe she could offer to buy her DF a six-pack every now and again for his Sunday drinks, cook dinner Saturday night, or any other small favour like that. Just to show appreciation. Because that's what adults and family members do to help each other out - they're all living in the same house. Although this isn't a reflection of the whole dynamic, it sounds a little bit tit for tat.
I wonder what the DD thinks of the whole thing?