Michael:
No budget, only because it's early days and I don't yet have a concept of what a budget needs to look like to have a meaningful discussion.
Are you thinking that the financial report would be changed from periods to some defined date ranges?
If you go to a 52 fiscal periods then isn't this more than just a financial reporter issue? Retrieve from sub-ledgers by 'period based on date' and selecting the period during GL batch entry come to mind as issues.
There would also need to be a 53 period solution. Again it's early days so we just been to figure it out, maybe this isn't that big of an issue. Thinking the problem through, in a 13 period cycle, every 7th year the 13th period gets 5 weeks rather than 4 (a 25% increase, because of time, when compared to the prior year). In a 52 period cycle, every 7th year, without a 53rd period, the 52nd period gets 2 weeks rather than 1 (a 100% increase, because of time, when compared to the prior year, which may not be acceptable).