The correct answer is: Will the programme finish?
The future tense is used to talk about things that will happen in the future. In this case, the speaker is asking about a specific time in the future when they will need to pick up their daughter from school. The most appropriate future tense form to use in this situation is the simple future tense, which is formed with the verb “will”.
The other options are incorrect because they use different tenses. Option A, “Does the programme finished?”, uses the present perfect tense, which is used to talk about things that have happened or been completed up to the present moment. Option B, “Has the programme finished?”, also uses the present perfect tense. Option C, “Shall the programme finishing?”, uses the future progressive tense, which is used to talk about things that will be happening or in progress at a specific time in the future.
In conclusion, the correct answer is: Will the programme finish?