The waiting period that Time referred to is not an indication of how long you'll be waiting for a decision on your entitlements. It refers to "waiting days".
For new Jobseeker claims, there can be 3 "waiting days" applied - this means that the first 3 days of the claim are not paid for.
There are no waiting days applied to Jobseeker's Allowance (JA) when going directly from Jobseeker's Benefit (JB). Entitlement will commence from day 1 of your JA claim.
You will be means-tested on parental income, so you will have to undergo a means test.
As you are transferring from JB, the age-related reduced payment for persons under 25 will not apply to you.
When you are 25, any means assessed against you because of your parent's income should cease and you should be entitled to full-rate JA, assuming there are no personal means assessed against you.
The SW Local Office should have already commenced the means assessment process for you, if your JB is about to run out. They may have asked you for the information when you made your JB claim several months ago. If they haven't, make sure to ask them what they need and bring it in as soon as possible. Means assessments can take several weeks.