Looking after grandchild

I wouldn't. It's not much fun being in someone else's house all day.
I might suggest a compromise in that you will go to theirs for the first few weeks until they have got used to it but then after that they will have to bring the child to you.
I was (early) retired when I looked after my grandchild two days a week, my DH was there too but we spent the next day recovering😊. It is nothing like looking after your own children and no way would I have spent my day off from work doing it.

Looked after all my Grandchildren and it’s much easier at their home.

If the terms on which you are providing free childcare don't suit you, you need to speak up and tell them how you want it to work. They can then either agree with you or make alternative plans. A certain level of empathy with the concerns of a new mum is reasonable, but you should guard against sacrificing your wishes entirely to that. If the arrangement is going to work, you need to be happy with it too, otherwise you'll become tired and resentful. Speak up now in good time or you'll find yourself rail roaded.

It’s so much easier to have the baby at their house, all their stuff is there. Cot, pram, high chair, food/snacks, toys etc etc. My mum always used to come to our house; she could happily take them out in the day too though.

She's asking you for quite a significant favour. You are entitled to agree do it on your own terms, you don't just have to agree to everything she demands.

"No sorry, that won't work for me. It's my only day off and I'll still have things to get done. I'd love to have baby but I'll need to collect them and take them home with me."

Reading between the lines is often not helpful.

Speak with them both and explain that you will delighted to help but that as it’s your day off you would like to be able to do it at home.
Also, ask if driving is a problem because if it is it’s going to cause issues. I have my dgc lots and they go everywhere with me whether we walk or go by car. If they have an issue with your driving you will be stuck to only walkable areas. Best to discuss now so that you all know what to expect.

She doesn't to trust you to drive because you had an accident a couple decades ago? But she wants you to drive an hour a day to be childcare?

That's weird.

I have done a lot of GC care. When they are little it can be good to be at their own home as everything is geared up for them there.

I think you should feel free to offer childcare on whatever terms suit you.

I think your DS and DIL should feel free to decide whether or not they want to take you up on that offer.

If they are unhappy with the terms that you are willing to offer, then you are free to decide that you can't offer childcare after all.

Neither side is necessarily wrong, it's just a question of whether you can find an arrangement that you're both willing to live with.

Logically it makes so much more sense to look after the baby in their house as they will have everything set up for the baby.

She obviously trusts you or they wouldn't have asked you to look after the baby so I think you're overthinking the car accident and lack of trust on this occasion.

I think you need to talk to her and address her fears, she will no doubt relax in time but initially it makes sense to keep baby in a routine at theirs doesn’t it?

How would she know about the accident. The only reason I can think of why it's better at her home is that everything will be there already. Why would you need trips out with a baby surely you would just go for walks with a pram. This wouldn't bother me at all.