Not British, but from what I read, don't the British people trust the NHS even more than government and police, and so any pushback against an NHS recommended app or approach might be minor.
The NHS is generally very well-respected, but it's important to remember that "the NHS" is really a large collection of distinct organisations and a huge number of people working within them, and to some extent the shared culture within that community. The professional ethics of those working in the NHS are generally respected, but that mostly refers to clinical staff and to some extent support staff. The management and bean-counters are not necessarily accorded the same trust and respect, nor should they be on the basis of past performance. The NHS has done some shady things with sensitive personal data on their watch, and we absolutely should be cautious of any cavalier attitude towards privacy if the NHS leadership and their political masters try to push out this sort of contact tracing app.
FWIW, I wouldn't install such an app if I didn't consider it to have sufficient safeguards, but then I'm a strong advocate of privacy and civil liberties who is also willing to give up various other "normal" parts of life to avoid compromising those principles in ways that make me uncomfortable. What should concern the powers that be in this case isn't me, but rather the number of my friends and family who typically do tolerate more intrusion into their privacy (for example, being willing to share a lot of personal information on major social networks) but have expressed concern about this app.
