Let's define better the situation and then analyze it:
Person A (you) : has £20K in the US.
Person B (a friend who is moving from the UK to the US for studies): has £20K in the UK.
Person C (a brother who will be starting studies in the UK): has nothing.
Person A (you) : has nothing.
Person B (a friend who is moving from the UK to the US for studies): has £20K in the US.
Person C (a brother who will be starting studies in the UK): £20K in the UK.
A sends £20K to B in the US.
B sends £20K to C in the UK.
B has the same amount of money, just in a different bank account, but
C changed states.
A now doesn't have money, and
C does, as the result of the transaction between
The gift tax issue I see is the transfer of money from
A (you) to
C (your brother). If you're a US tax resident then you have $14K exemption from gift tax per person per year. £20K is more than that, so it will be subject to the tax.
The fact that a third person was involved as an intermediary is irrelevant - for the purpose of gift tax there's no distinction between using a bank for transfers or a private party.
Keep in mind that paying tuition directly to the institution on behalf of your brother may help you mitigate your gift tax liability - tuition payment made on behalf of your brother is exempt from gift tax. But it has to be made directly to the institution, it cannot pass through your brother.