difficult child 3 does this, constantly. And I can assure you - getting angry only makes things worse. He HAS to finish what he is saying (husband & easy child 2/difficult child 2 are the same there) and it's no use getting embarrassed, or angry, or fed up - just shrug your shoulders and introduce yourself. Because frankly - there is nothing else you can do.
At other times I talk to difficult child 3 and we sometimes practice more appropriate ways to meet & greet people, but AT the moment is never the right time to say, "Ok, kid, time to rehearse." It's too late then, you're in performance mode and not rehearsal mode.
difficult child 3's best friend is doing this too, his mother said. He tries to begin a conversation with other kids but is using his football cards as a conversation starter, walking up to kids and shoving his cards right in their face and talking non-stop about everything he knows on his football heroes. Other kids don't like it, they think he's weird.
difficult child 3 had a speech pathology assessment about two weeks ago. The report came in yesterday and includes information on how he gives too much information; he has no inner self-censorship. It seems to connect to his need to connect socially, plus his difficulties with memory recall (which links in to how his brain stores memory and cross-connects it). He WANTS to please people and so he shares EVERYTHING. If you ask him what he had for breakfast, you're likely to get absolutely everything breakfast-connected, plus any other information which even slightly cross-connects (ie which he thinks of as he's talking) and EVERYTHING comes out. He's got enough savvy to not tell you his computer passwords, but that's maybe because he's a bit older now.
With difficult child 3, and his friend, it's a part of their autism. It connects to inappropriate socialisation. And if you try to slow them down or direct them, you get tantrums PLUS you get blamed for their social failure in that situation. "Mum, why did you interfere? Of course that kid won't talk to me now, because you interrupted me just when I was getting on fine."
In "Explosive Child" parlance, this is somewhere between Basket B & Basket C. You can't help them change if their brain is not yet mature enough to recognise there is a problem; and you can't make them change if your efforts clash with other issues (such as a need to seem independent; a need to be allowed to finish everything they're saying without interruption (because it takes very little interruption for them to lose their train of thought - and this connects, again, to how they lay down language memory).
It's complex, which also makes it harder to work with.
Before you label it as pressured speech, double-check exactly what is meant by pressured speech. I remember trying to apply that label to difficult child 3 and finding that it superficially resembled it but was not, in reality. You can have non-stop, unable-to-be-interrupted speech that isn't necessarily excessively fast. Pressured speech is typically extremely rapid; you can't interrupt it because you can't even get a syllable in edgewise. With difficult child 3, husband & easy child 2/difficult child 2 there can be pauses, but woe betide you if you assume the pause means you may now speak - they're just getting the next idea, or taking a breath. If you cut in, they lose the thread and then get angry that they've forgotten what they were talking about.
However, they can and do get off topic really easily, because part of the memory problem that causes this also means that as they produce the verbal output, they also can distract themselves from the original topic. Again, husband & easy child 2/difficult child 2 aren't too bad with this, they are adults and have adapted. When they are tired or not well, however, they are worse at losing track.
Pressured speech is a symptom of bipolar, schizophrenia and anxiety. However, what we observe (according to the report on difficult child 3) is language-based and is connected to their autism. But it might take an expert to truly confirm the difference.
Back to what you can do - about all you CAN do, is rehearse. Practice between the two of you, set up fantasy role-play scenarios and make a fun game of it. Maybe you can play her being inappropriate, and let her play the new friend. Maybe if you can videotape your games and play them back, it can help. Or use examples in film and TV (comedy is full of people being socially inappropriate, it is what makes comedy so often).
Another suggestion I've made - if the problem is similar to difficult child 3's so it's connected to language and how it connects within the brain, then a really good game is to get one of those 20Q games. They're a lot of fun anyway, and can train the brain to increase the connections concerning whatever it is they're thinking of. For example, you're thinking of "apple" and the 20Q asks questions like, "Is it bigger than a house?" So you immediately visualise an apple, and a house, and make the assessment. For some kids, depending on how their brain initially laid down language, these connections haven't been properly made and their verbal diarrhoea is the result - the game helps. Besides, think how much fun you'll have playing with it when she goes to bed!
Good luck. It takes time, a lot of patience and rhino skin a mile thick.
Marg