Trains to Oxford take 60–70 min (£5–£32) from London's Paddington Station
Buses to Oxford take 100–110 min from London (£1–£15)
Coach transfers to Oxford from Heathrow take 80–90 min (£23), from Gatwick 2–2.5 hr (£28)
The drive west from London to Oxford takes about 105–120 min.
The Queen doesn't need a passport. Since all British passports are issued "In the name of Her Majesty," the Queen can just vouch for herself—though her husband Philip, son Prince Charles, and everyone one else in the household do have passports.