What grants US Presidents the power to give Presidential Pardons, and why can't (or don't) UK Prime-ministers do it?
The US President's Pardon-Power
The United States Constitution gives the President the ability to pardon criminal offences:
he shall have Power to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment [Source: Article II, Section 2.1]
Similar Powers in the United Kingdom
In the United Kingdom (and likely other Commonwealth nations) the monarch (or Governor-General, on their behalf) can pardon crimes through the royal prerogative of mercy.
As with many other powers of the royalty, in today's world it is only exercised at the request of the appropriate minister. In this case, it is the Secretary of State for Justice (or Defence, in military cases).