If you didn't see the episode, it is (theoretically) available on the ITV website. That said, ITVPlayer is having trouble at the moment, and it kept cycling through all the "sponsored by Haven" promos rather than showing me the episode. Anyway, I need to give a brief synopsis to explain the context, so you might want to watch it on your own before I spoil it.
The basic premise was that someone touched a fungus: this spread all over his body, and turned him into a plant monster. Similarly, if anyone else came into contact with the spores (e.g. by touching the monster) they would suffer the same fate. The ARC team tried various methods to subdue the creature, but they discovered that bullets weren't effective. To quote a line from Swamp Thing: "You can't kill a vegetable by shooting it in the head." Eventually they figured out that fungi like damp warm conditions, so it would die in a cold (sub-zero) environment.
Towards the end of the episode, Jenny (team leader) was stuck in a cold room with the monster, and by the time the monster froze it had already touched her. This meant that the team couldn't take her out of the room until the cold had killed off the spores on her skin. Once her face was clear, they rushed her out and resuscitated her.
So, where did they go wrong?
1. As a first aider, the first step in my primary survey is "Check for danger". In this case, ignore the sci-fi monster stuff: Jenny had a toxic substance on her skin, which would kill anyone who touched it. It would therefore be sensible to wear gloves! That's standard practice for first aid treatment anyway (mainly to protect the casualty from infection if they have open wounds), and since the rest of the team were bundled up in heavy coats it would make sense to wear gloves to stop their own hands from freezing.
2. They saw that the fungus had disappeared from her face. How about the rest of her body? She was non-responsive by this point, so she couldn't give them any useful information. Thinking back to my ETA training course, this is where I'd use Tuff-Cuts to remove the rest of her clothing, to do a secondary survey. (The idea is that if someone has been stabbed, and they've got blood gushing out of a wound on their back, you need to find out about it as soon as possible so that you can deal with it.)
3. Abby used a bag-valve-mask. This is basically an improved version of mouth-to-mouth resuscitation, where the mask fits over the casualty's mouth and nose. You can use the mask on its own, and blow into the top: this avoids the "icky" factor of touching them directly (particularly if they've vomited), and it can help you get a better seal. With the bag attached, you squeeze that instead of blowing air into it. (You can also attach an oxygen cylinder, but you still need to squeeze the bag.)
The key point here is the seal: you need to make sure that the air inflates the casualty's lungs, and you do that by making sure that it has nowhere else to go. For instance, with mouth-to-mouth rescue breaths, you have to pinch the casualty's nose, otherwise the air will just come straight back out. Also, think about what you do when you blow your nose: you use the airflow to clear out all the snot that's in there. So, blowing air into the mouth and having that shooting out of the nose at you wouldn't be a good thing!
The mask helps, but you need a firm grip (known as an "anaesthetist's grip"), otherwise the air will escape through the gap between the mask and the skin. When I've practiced on "annies" I've sometimes found that the chest doesn't rise, even though I'm pushing down on the mask, because I've left a tiny gap somewhere. Coming back to Abby's approach, she held the bag in her right hand, balanced the mask on Jenny's face, and used her left hand to feel for a pulse. That ain't gonna work! Checking for a pulse is a waste of time anyway; if you know the casualty isn't breathing, you can assume that they don't have a pulse. So, Abby would have been much better off using her left hand to hold the mask in place. In fact, since there were at least three people in the room who were standing around watching, the ideal approach would be to hold the mask with both hands and get someone else to squeeze the bag.
4. Nobody even attempted to do chest compressions, so they weren't actually doing CPR. As a related issue, the chain of survival has "early CPR" followed by "early AED", and nobody attempted that either. Again, what they should have done was to remove layers of clothing: that would give them access to her chest (to attach the AED pads), and it would also make it easier to see whether her chest was rising (i.e. whether the BVM was effective).
At a meta level, it's curious that the first series of Primeval had some pretty flimsy justifications for the female characters to wander around in their underwear. Now that it would actually make a lot of sense, they don't do it!
5. Nobody called 999. They should have done that as soon as they knew that she wasn't breathing; if they actually succeeded in resuscitating her, they could always call back to cancel the ambulance, but realistically she would still need to go to hospital anyway. It's possible that they didn't want random people coming into their high security facility, but if necessary they could arrange for the ambulance to meet them outside the front door.
I'm not an assessor, but if I acted like that in my assessment then I'd expect to be "referred" (i.e. they'd fail me). In fairness to the characters, none of them were trained medics (unlike "Slaughterhouse Jones" in Doctor Who), so I'll cut them some slack for botching the job. However, I do think it's worthwhile to draw attention to mistakes like this, in case viewers encounter a similar situation in real life.