Present tense and conditional tense in a stated past tense

Solution 1:

In your A sentences we have a straightforward situation. Backshifting after a past-tense reporting verb is nearly always correct, even when the statement is a universal truth, so A2 is correct.

Non-backshifting, if the reported statement is still true or (as in first conditional sentences) possible at the time of reporting is also correct, so A1 is correct, as is B1.

B2 is inconsistent. "He just told me that it's important and to give it to you when the time was right." The speaker has started with a non-backshifted 's (for is) and then has gone on with a backshifted was. This would almost certainly go unnoticed in speech. We often start an utterance with no clear idea of how we are going to complete it, and the conflation of two different ideas is common*. In writing, where we have more time to think and correct, B2 would be corrected by a careful writer so that the reported statement would have either two present-tense forms or two past-tense forms.

For example "I have seen him two weeks ago". Here the speaker has conflated the ideas of "I have seen him" and "I saw him two weeks ago and said what no careful native speaker would write.