Bath vs NYU for Mech Eng

Hi, I’m an international student and received offers from Bath (with a placement year) and NYU Tandon for Mechanical Engineering. I know that NYU is a prestigious university and has a very low acceptance rate but I don’t know much about Tandon’s academic reputation. Since NYU’s engineering programs are ranked lower than the other degrees, it makes me wonder if NYU’s reputation stands for Tandon too. Also Bath is considered very successful in its engineering degrees and ranked top 5 in the UK rankings but somehow ranked around 100-200 in world rankings.

Additionally, I hope to pursue an MBA after graduation and work towards becoming a manager. As you can imagine, I’m really struggling to make a decision, so I would truly appreciate any help or advice you can offer!

Given that you're aware of the distinction between a university's reputation / prestige generally and for a specific subject area, I'm surprised that you haven't checked these two universities rankings in the subject area concerned.
According to QS World University Rankings, for Mechanical, Aeronautical & Manufacturing Engineering (source), Bath ranks above NYU:

Hi, thanks for the response!
Of course, I’ve checked the QS rankings for Mechanical Engineering, but according to the Times Higher Education rankings, NYU is ranked over 100 places higher than Bath. What really confuses me is that I’m not sure which ranking to rely on — QS, THE, Complete University Guide, or The Guardian.

And therein lies the problem with rankings!
They all use different metrics, and weight those metrics differently. If you're going to use rankings for anything other than to get a rough idea of a uni, then you need to look at the methodology they've used (metrics and weightings) and then focus only only the metrics which are important to you - not the compiler of the table.
Your post seemed to focus on international reputation and prestige ("NYU is a prestigious university", "Tandon’s academic reputation", "NYU’s reputation", etc.). If that is what's important to you, then focus-in on just those metrics within the various league tables.

Hi, I’m an international student and received offers from Bath (with a placement year) and NYU Tandon for Mechanical Engineering. I know that NYU is a prestigious university and has a very low acceptance rate but I don’t know much about Tandon’s academic reputation. Since NYU’s engineering programs are ranked lower than the other degrees, it makes me wonder if NYU’s reputation stands for Tandon too. Also Bath is considered very successful in its engineering degrees and ranked top 5 in the UK rankings but somehow ranked around 100-200 in world rankings.
Additionally, I hope to pursue an MBA after graduation and work towards becoming a manager. As you can imagine, I’m really struggling to make a decision, so I would truly appreciate any help or advice you can offer!

NYU globally certainly has the stronger reputation, im not sure its an engineering powerhouse, I can’t recall ever coming across NYU contributions in any engineering journals. Id say it certainly would probably be the best university for a career in the US, and probably across the Middle east/Asia with its brand.
For a career in the UK id say for engineering industries Bath would be the better route to go, it is very strong here, and in certain industries has great engineering research bringing with it industry connections…
Outside of engineering particularly financial services id still say NYU is better for UK careers.
For an MBA, it’s really too early to say if that is appropriate. Standard MBAs really not needed or desirable for junior management positions in technology & engineering companies, with a much stronger emphasis on technical & leadership qualities over an MBA which focuses on strategic & fiscal decision making and this is far more suitable for professional services industries. What you tend to see is exec-MBAs, these are done by potential C-suite candidates typically in there 40s after having already climbed through middle management and are typically company sponsored. And again not something you can really plan for at your stage as it would depend on your career path as to if this is useful.

Thanks for your detailed and kind reply. Especially your explanation of the difference between standard and executive MBAs made a lot of sense and gave me a clearer picture of what to expect long term.
I also wanted to ask: is there anything similar to a management minor in the UK system (for Bath), like an elective course or certificate that can be taken alongside a Mech Eng degree?

There are masters degrees called engineering management, typically one year masters degrees however the focus on project management not being a manager.
In my experience companies wouldn’t seriously consider management education of any value in early career hires anyway, typically management training is given when someone is viewed as a potential line manager or just been promoted and is typically one or two day courses through professional CPD courses. Additionally large multinational companies typically have there own management training which is like half a day per month for new managers.
In engineering and other technology focused industries, managers are generally leaders who can solve technical problem’s, deliver projects on time and in budget and foster a culture that enables others to do similar. Management training is not really needed and more focusing on having excellent technical and people skills.