This sounds like the end goal of a combination of items already in development within NHSD I'm working within NHSD at the moment within Product Development and will pass your story and idea on if you don't mind. Below are just some of my thoguhts though and links to items that are currently avialble or being worked on that could eventually help with the problems you outlined.

NRL aim to create pointers and provide access through to data in disperate data stores. Currently supports document stores but plan is to extend this for other data types. https://digital.nhs.uk/services/national-record-locator

NBS The national booking service looks to have been rolled up for COVID but the intent is having a place to book and manage appointments for procedures you are recomended to have (intially focused on vacinations). https://digital.nhs.uk/coronavirus/vaccinations/booking-syst...

Then you have BARS that is focused on standardising disperate booking formats so that they could all then be usefully referenced using NRL possibly then linked to NBS. https://digital.nhs.uk/services/booking-and-referral-standar...

I'm not directly involved but is some details of the breast screening applications here, I suspect the above would need to be connected up with this to all work together. https://digital.nhs.uk/services/screening-services/breast-sc...

I know the call and recall service is currently being replaced for cervical screening with CSMS (Cervical Screening Management system) so I would suspect that if its not already on the todo list that a replacement for breast screening would line up based on the success of that. Much of the aims of this service are to solve some of the problems that you mention but for cervical screening. https://digital.nhs.uk/services/screening-services/national-...

All of the above links will have some form of contact details also if you would like to contact the various teams.