Yele - I know exactly what you mean, I received the letter with the new codes, and I changed the numbers with my bank for my Faster on-line payments. I then spent an anxious few days hoping the payment got to the right account - which it did. However, like you, I am still receiving statements with the old sort code and account number on it - but I am definitely sending payments to the new numbers and they are getting through to my account.
All I can suggest is that you re-check your on-line banking to make sure that every single number on the new details is correct on your record - and if it is, send a test payment of say £1 to see if that gets through OK. As I recall the original letter said that payments sent to the old account details would continue to get through to accounts for a few weeks yet.
I'd still try ringing Tesco to see if they will consider clemency during this changeover period.