In short, I don't necessarily see a scenario where police and TT would get into armed conflict. Police shoot someone down who has TT services? Police can just treat them as paramedics IRL; let them come, fix them up and then take them into custody, whether from the location or hospital later.
Ha! Not in Night City. NCPD don't generally give two craps. They are quite corrupt, with the exception of Harbour Police (clean) and CSWAT (nuts).
Unless they have an interest in you surviving, they aren't going to bother saving you or fixing you themselves. And TT isn't going to give you up either.
So if you are in armed conflict with the cops, ( not unusual for Edgerunners) and you get h8urt and have a TT contract, yeah, TT will come.
What happens next varies. TT is -going- to get you, that's the contract. Sometimes they'll take a pass, but super-rare. Their business model would go to poop. If you are in a really bad place like Arasaka tower and/or deep in Combat Zone, maybe. Or if it's a MAX-TAC op.
Otherwise it's the paramedics call. Often they will swoop in and everyone gets out of their way. Including NCPD. If not, it's machine-gun time. NCPD can always arrest you at the hospital.
Unusual for the cops to bother to get in TT way - no percentage in it. Corrupt, remember? And if TT can, they'll let the firefight die down and resuscitate the intact. If they can't wait, then blammo, in they come. As hard and fast as necessary.
As for the cops going after Trauma Team, how? Arrest them? Who? The board? They might sue them or press charges against the crew ( if they can find out who it was), but that's what TT has lawyers for.
NCPD is a municipal police force - and not a fantastic one. TT is a megacorp. That's how that works out.
Oh, side note that I feel a lot of people aren't aware of: in 2020 at least, Trauma Team doesn't have hospital facilities. They just keep you alive and get you to a hospital. Where, again, the cops can arrest you if they care to bother. And can find you.