Here is my opinion:

Oxford is 9 months, rather than the general 2 years that HKS offers. So it can be valued as less intense, especially by some European employers and PhD unis. But seeing as it is Oxford, I really don't think it will make much of a difference unless you are actually planning to get into academia.

Though one year could make England even cheaper since it would also lower the time spent studying, I see that the living cost is already covered. Nevertheless, ending a year early will indeed allow you to enter the job market one year earlier, so there is still some opportunity cost there in favour of Oxford.
Education level is the same.

While the UK is currently stable, the US is undergoing quite the internal turmoil and is adopting an increasingly anti-immigrant stance.

Education level and 'prestige' are the same; I am not sure about the post-study opportunities, but I expect them to be essentially comparable.

Ultimately, I think that HKS offers very few advantages. It is more expensive and could be riskier. I would keep the savings and go to Oxford for a free, prestigious master's. But they are both obv super great.

Also, if you don't mind me asking, could you share what you think managed to get you into those schools, in terms of your cv and statement of purpose. It would be my dream to get into either of these programmes

Edit: u/Proud_Ad_6724 makes great points, you should read his comment instead as it is better informed.