But GDPR dictates that personal info cannot be given without consent
No it doesn't, it's simply without appropriate grounds, and public health is of course a grounds - equally that doesn't mean any use of data you can just say is public health, but certainly the information required to trace someone in a pandemic situation seems likely to be reasonable.
Why not the google/apple app?
'cos it's flawed in a different way, it's about the proximity of phones, and the proximity of phones is different to the chance of virus - if a workman sits in a hole outside your house repairing the phone lines for 30 minutes, he'll probably be a "close contact" for phone purposes, but the chance of a virus transfer would be almost nil.
Because of that, you won't actually know if a "G/A" report of a contact is a real one, and there's no-way to actually find out - unless you're one of the people who deliberately de-anonymises your contacts (which you can do with multiple phones and can therefore tell which one is triggered).
This will cause either unreasonable fear and lockdown of you because of contacts that couldn't have done it - what if the only person you though you'd visited was your Gran? It's also much less effective to only be told by a machine that you need to isolate without any context - "You spent an hour in that McD's sat next to your mate John who's just tested positive" is considerably more persuasive than "You've had a contact, we can't tell you when or where as it's all anonymous"
The when is also relevant - going back to the Gran, if you saw your Gran on Monday, knowing that you only had the contact on Tuesday is really important for your fear for your relative, the G/A system makes that more complicated.
Hiding from track and trace is probably okay, you don't need to track everyone to keep the transmission low enough to stop spread, just track most, and social contacts will get most people that matter even if they don't explicitly get you.
But yes, it would've been helpful if the government wasn't so grossly incompetent, but then we got what we voted for.