The Captain will have signed an estimated or projected loadsheet. The take-off performance will have been worked out based on these figures with a margin for error. The crew will not proceed with the take-off until the correct figures are passed by voice radio or ACARS (and acknowledged by ACARS). As long as they are within the allowed margin of error, the previously worked out take-off performance is used, or if outside the margin, which is quite rare, the take-off performance will be recalculated.
Now, you asked a sensible question about CLC and then added your personal opinion without actually knowing anything about it! It happens to work fine. I have been using it for some 12 years with no problem (apart from the figures sometimes fail to come through on ACARS or are delayed). It allows the airline to save those vital 10 minutes+ when someone would be frantically entering a load of numbers in a computer to try and get a printed loadsheet, rush to the aeroplane and get the crew to check and sign, then close the doors and kiss them goodbye at last! This time could then be taken up by actually pushing back and getting going. You obviously won't believe this as your question was so prejudiced, but it works, and works well, and BA is by no means the only airline doing this. It satisfies all requirements regarding legality, and I believe, as a former BA pilot, is as safe as the previous system.