I’m not so certain that there is a fixed British way of saying or even writing dates. For the date in question, I would use, interchangeably:
The twenty-second of January, two thousand and three.
or
January the twenty-second, two thousand and three.
I also admit to writing dates in either format: 22nd January 2003 or January 22nd 2003.
In both cases, I also acknowledge Stargzer’s comments of 22 Jan 03, inasmuch as the "and" in the year number would often be slurred though never completely dropped.
The real problem with US/Brit dates comes when the month is written as a number. The US format of mm/dd/yyyy and Brit (and most of the rest of Europe) use of dd/mm/yyyy can and does frequently lead to misunderstanding and confusion.
Faced with the date 1/4/2003, I would pronounce it:
The first of April, two thousand and three.
whereas I would expect the US pronunciation to be:
January fourth, two thousand three.
The only country that I know of where dates are normally written with the least chance of confusion is Sweden, where they use yyyy-mm-dd. But maybe even then a US-minded reader might assume it is to be read as yyyy-dd-mm?
An anecdote (completely true):
A couple of years ago I spent some time working in one of our US offices. I was discussing this issue of date pronunciation with an American colleague. He was adamant that the US way of saying dates was 100% consistent, unlike the British approach as mentioned above. My question to him was "On what date do you celebrate the Fourth of July?", his instant reply was "July fourth"!