No - because it will be absolutely useless.
I work in a school. I caught something that the NHS says was Covid-19 in March from having close contact with multiple sick teenagers. A large number of other staff were also ill with something over the last couple of weeks.
The reason why the app would be pointless is that the certain vectors of infection - sick teenagers - do not have their phones switched on in school. Staff are also not allowed to have their phones with them. The best part of 2000 people in a smallish space for up to ten hours a day, and not a single Bluetooth handshake between them.
When I worked in the NHS, we weren't allowed our phones with us.
When I worked in a callcentre, we weren't allowed our phones.
When I worked on a supermarket checkout, we weren't allowed our phones.
DP isn't allowed his phone at work. 300 clients a day pass through his workplace, all spending at least an hour there in close contact with the staff.
Apart from grumbling about how shit women's and girls' clothes are that you don't have pockets to hide your phone in, apart from grumbling that all it would take is for one kid to report they had symptoms to scare a teacher/get out of detention/avoid homework being collected, the app doesn't allow for the vast numbers of contacts that aren't able to use phones.
The apps simply won't work for contact tracing. Great for information gathering on adults' movements and behaviour, but useless for the declared purpose that it's been sold to the government. If it was a compulsory push app to all phones, it would probably end up, as Oyster cards did, handy for providing evidence of where somebody goes and who they met up with for criminal prosecutions.
But actually providing an accurate picture of disease spread? No chance.