If you want FOOD then Borough Market is the place to go (as already mentioned bear in mind any customs regulations at your destination of you want to buy things to take with you though many of the stores do sell pre-packed items with labels of ingredients on them)

If you want PRESENTS then go to Covent Garden.

As for the tube station whether it is open or not just avoid it as it just gets very busy. Far better to use Leicester Square or Holborn. Yes it's a longer walk but it will be less pressured.

BTW last year I heard an American saying how 'neat' it was that they had named the Apple Market at Covent Market after the electronics store!

Mind another one asked at Borough Market why they build it under the railway bridge when it was the other way around