Problem 1
If she has the baby in the UK, she is liable for NHS fees. Currently she is financially supported by her Mummy and Daddy. When they find out, God knows what they will do in terms of financial support. I am betting she won't tell them until the pregnancy is well advanced to prevent them from using finances as a way to get her to terminate.
Problem 2
At some point she will have to tell her parents for financial reasons, due to above point. The DS asking if he can move in with gf and baby suggests that she knows she's going to get cut off financially and her parents will no longer support her accommodation financially.
Problem 3
She will not have a visa to stay if she drops out of education. There are a few issues with this. No visa = may not be able to rent accommodation. OP needs to know the visa status if this woman decides to overstay her visa and OP provides accommodation. There may be legal implications.
Problem 4
No visa = problems financially generally. This woman will be unable to work. The government may not be able to deport her due to human rights considerations but it can force her into a desperate situation where she has no choice but to leave because she has no job and no accommodation. She will not be able to take the baby if the OPs son objects. This isn't an unrealistic scenario, especially if she and son fall out. She'd be dependent on the son for finances / accommodation so if he no longer is willing to support her with this, only the baby, she's screwed. People don't understand that the British government don't have to force anything in most cases for this reason. She second she leaves she won't be able to come back too.
Problem 5
Having a baby to support makes it harder for her to apply for a visa on her own terms because she financially will have to pay fees and provide childcare on top of working. Again, this is another reason the son might be asking to move in. It's for free childcare. She will have to be able to get a good enough job to meet the criteria for entry. She's educated enough to have that potential, but given how hard it is for British graduates to get a job ATM don't bet on her being able to do this. There's often extra paperwork for employers so unless she's exceptional she may struggle to find the necessary employment.
Problem 6
If she does leave and takes the baby, what is the OPs son going to do? He can object and get the baby back if he does it immediately (as UK will be the baby's residence to that point). If he lets go and objects much later on he may find he's shut out because south Korea becomes the child's place of residency.
Problem 7
If she is financially cut off by parents and financially becomes screwed her by visa limitations and a falling out with boyfriend, her only option may be to return to South Korea alone in order to support herself financially. This may have financial implications for the son as he may be left holding the baby with no financial support from her. If she goes 'temporarily' she may find unable to return because she can't get a work/residency visa. Especially if she overstays her current visa (if she doesn't this she'll have trouble coming to the UK on a tourist trip as she would have to apply for a visa for that and wouldn't be allowed to come on a visa waiver basis - noting the additional cost in getting said visa).
Problem 8
All these financial obligations have an impact on the son's ability to stay in education. He may just graduate before the baby is born but only just. Either way there's some pressure to get a job quickly at a time when graduates are struggling to get employment. He will automatically have planned to move home if he hasn't got employment. OP has now put foot down on this on terms of gf and baby. He will have to provide for gf and baby who may find themselves unable to rent due to visa issues. You have a situation of the gf being potentially homeless, with new baby. With no visa she's got a problem - she isn't going to be eligible for accommodation and she can't work. Even if she does get a visa and can work shed have to pay for childcare and accommodation. So who realistically is going to end up holding the baby? Probably the son (especially if he has no job) or the OP.
Problem 9
If the gf doesn't get a work visa she has to apply for a spouse visa. At present this isn't an option as you have to prove a long term relationship and they simply haven't been together long enough. That's years she will either have to return to South Korea or overstay and be financially supported. Whilst she can't be deported for human rights considerations, I'd be mindful of the political climate - this human right isn't guaranteed to stay. We have Reform threatening to leave these human rights conventions and as it stands the polling suggests this may be a realistic scenario. In which case law changes may mean she IS deported at some point.
Problem 10
Say they do make a go of it somehow over coming temporary visa issues and financial issues, they STILL will have to save for permanent residency for her. That's a lot of money.
Problem 11
Moving to South Korea isn't a viable option for the son. He won't be eligible for certain visas which are normal for young people due to having a dependent child and he doesn't speak the language.
Problem 12
Will the son step up? If she struggles with visas and finances and he isn't providing, there will be a temptation to fuck off back home to South Korea. If he doesn't object and she takes the baby and settles there with the baby he is going to find he won't get the baby back. So he needs to do some growing up in terms of whether he wants to be a dad or not because this is a realistic scenario. He will need to consider employment and childcare issues if he does object too.
My point being in pretty much every scenario, realistically the OP gets dragged into matters because of visa and financial considerations. There is a huge chance of family separation for practical financial reasons even if the gf isn't deported. At the heart of this someone is going to end up holding the baby. By the sound of it, it very much sounds like the son tried to pin that on mum right now in his initial thoughts and the OP is perfectly sensible in saying "no, you need to work this out yourself" at this stage if only to get him to start realising the mess he's in and what the implications are.
Other posters pointing out the OP has a child very young miss the point that as a single mum here, she had the right to work / had access to benefits - these are not things the gf has and this has significant implications for the son. The son does have the right to these - the potential for him to be a stay at home single dad on benefits is real. The OP has the right not to get involved.
Realistically the OP is going to face some difficult financial and emotional choices herself over this I suspect. She may end up with a grandchild in another country she never sees. Or having to take on childcare or financial burdens because of her feckless son.
The fact the son isn't talking to his mum because she said no, just highlights his immaturity and he needs to wind his neck in and apologise. The chances are he is going to NEED his mum at some point in this because of the mess he's made. If he manages to sort the visa and financial situation without his Mum, great; Mum has done her job as mum by saying no and forcing that. Unfortunately given the barriers here, that's actually an unlikely scenario so he should be sucking up big time to Mum.
And Mum is still well within her rights to say, I'm not prepared to do this because if she does say yes, the son may merely take her for granted and treat her like a doormat.
Realistically the chances of this relationship surviving are limited and that in its self means there's heartbreak for all these somewhere down the line. OPs job as mum is to point this out to feckless son because he's got himself into a terrible mess because he needs to know in order to avoid certain pitfalls. He might feel his mum is being unsupportive but frankly that's tough shit - he needs a very rude awakening. The pregnancy test just shows he's living in online la la land and doesn't live in the real world.