Okay. To my knowledge, the app works out if you've been within 2m of someone for 15 minutes who later tests positive.
covid19.nhs.uk/risk-scoring-algorithm.html
If you've been to M&S then, unless you were standing next to the fruit counter with the same person for 15 minutes, then it shouldn't trigger the app.
Also, the app doesn't understand risk profiles. Hence, if I stand 1.9 metres from a friend who later tests positive for COVID-19 while we're both wearing FP2 masks outside on a very windy day, then the likelihood the other person will get COVID is pretty low. In contrast, if we're both standing in a stuffy office unmasked, or I'm a dentist doing a root canal and the COVID-19 patient is my patient, I'm probably at very high risk.
I expect the person who got a notice to self-isolate when delivering a food parcel to a friend with COVID was within 2m of someone at the other side of a closed door.
In short, given what we now know about how COVID spreads, and the places you've been in the last week, I would have no concerns whatsoever about collecting my kids from school in a car, and not getting out of the car. You are, after all, unlikely to have caught the disease and going to be at home with the children subsequently.
My suspicion is it's either a false notification (at which point, you're exposing yourself to a risk of COVID by having your kids in a taxi with a stranger), or someone in the supermarket had COVID. Given you were, presumably, not close to them for an extended period, you may be unlucky enough to get the disease if there was extensive aerosol transmission, but it's also likely you didn't get exposed.