The ideal way to do this would be to take her to a vet as soon as you can when they open, and get them to cut this nail out. If it has gone into the pad....and you say this only happened a day or so ago?...then I guess it hasn't had much time to grow in there. But it is obviously causing great irritation and as you mention, the pad is starting to swell.
It shouldn't cost very much, if the case is simple, with no other complications such as infection. If that were the case then it would also involve at least a course of antibiotics, and possibly wound drainage, dressing, check up afterwards...etc. That will cost more.
But of course, even if all they do is clip the nail they will probably (rightly) give you a bottle of antiseptic also. And there will be the consultation fee. That still shouldn't amount to much more than (at a guess) 50-60 pounds. Probably less.
Advice for home treating a dog from a stranger, over the internet, who doesn't know the case, or the dog, is not wise.
The best person to gauge what is happening to her foot is a vet.
But even though I have said that you could try clipping the nail yourself using some good nail clippers suitable for dogs' claws. So that the embedded part is set free. Do not clip too high. Just release it. And gently pull out the embedded part of the nail.
Yes she won't like it as it will hurt. You need someone to hold her very still while you do this.
Then examine her foot. Is there infection? Is there a hole? How swollen is the pad? It is up to you to use your judgement about how bad that wound is. But remember that any puncture wound can spread infection deeper into tissues than a cut or surface abrasion.
You could bathe her pad with an antiseptic, and keep a VERY close eye on what is going on! Any signs of it not starting to heal in a day or two, and PLEASE get her to the vet.
She may get dirt in the wound as she will have to go out for at least poop/pee. So that means every time she does, gently clean the wound with antiseptic each time.
Restrict her walking until the foot is healed.
Unless advised to by the vet, don't dress the foot. but make sure the wound is kept clean.
Or you could put a little sock on her foot when she goes out to do her business. You will have to anchor the little sock with a string or ribbon which goes around her shoulder across the other side. Take it off when she comes back in.
The smell/taste of the antiseptic hopefully will stop her licking at that pad.
One thing you do not want is a spreading infection. Or an eventual amputation because of bone involvement!