When you change your statement cycles to a later date would you first need to send your statement on the 15th and then another on the 20th (if you were changing from the 15th to the 20th or another date).

Yes, if there is electronic funds transfers occuring. See Reg E,
Sec. 205.9 Receipts at electronic terminals; periodic statements.
(b) Periodic statements. For an account to or from which electronic fund transfers can be made, a financial institution shall send a periodic statement for each monthly cycle in which an electronic fund transfer has occurred

From the Reg E Commentary, it sounds like a one time variance to establish a new cycle date would be permissible even if the result would be a cycle that was not "reasonably equal". So to answer your question for your example, it does not seem like it would be necessary to send 2 statements within the same month.

9(b) Periodic Statements

1. Periodic cycles. Periodic statements may be sent on a cycle that is shorter than monthly. The statements must correspond to periodic cycles that are reasonably equal, that is, do not vary by more than four days from the regular cycle. The requirement of reasonably equal cycles does not apply when an institution changes cycles for operational or other reasons, such as to establish a new statement day or date.