It's possible that the detail in the new legislation will force rescues to revisit this issue. My own view is that the owner's details should be on the chip register, rather than the rescue. I've said owner deliberately because most rescues don't want to fall under pet shop legislation or register as dog wholesalers, so there's this waggle dance of adoption fees and loans rather than a transfer of ownership. In practice, you, the adopter are the legal owner, and if you incur a FPN for dog fouling, or your dog bites, it will be you who's responsible not the rescue. Besides, if your dog is lost, it's important that whoever scans the dog can get it back to you promptly. Not all rescues have 24/7 contact (very far from it, in fact) so again, it's important that the owner's details are there (and it's always worth repeating that you should check and update these details regularly).
Also, if the rescue details are on the chip, it might mean the dog has a place to go, but it won't necessarily make it any easier for them to reunite the dog with you, especially if you have moved house.
Our own experience with our dogs (rescue greyhounds) is that in several cases the rescue details were wildly inaccurate, and sadly files were kept in the home of a volunteer who subsequently left the organisation without returning them. Fortunately we were able to check and correct details but this is not an unusual occurrence. This is for a long standing rescue which is well known, not a one man band. Rescues also sadly close, or become inactive.
For one of our other dogs, the file with the details is kept in the home of the rescue organiser. If anything happens to their home (fire, flood etc) the records go too.
This is not to point the finger at any particular rescue, but I've not yet seen many rescues take the issue seriously of data management and security.
It's a different picture for foster dogs - will be good to see any detail on how that will be handled.
If anything happens to our dogs, they are chipped, the greys have tattoos, they wear collars with ID and we have pre-registered them with DogLost. I also carry a card in my purse with emergency contact details for caring for the dogs, and if anything happened I'd be on social media sharing their details like mad.