Thanks @DenverChris for coming to the forum to seek clarification. Just to make sure everyone is on the same page I just wanted to empathize what should happen under these circumstances.
Tower should always keep pattern work (because the aircraft are flying a VFR pattern). If the tower controller does not deem pattern work acceptable, which could be due to weather below VFR minima or if they are too busy (as we see on FNF etc), then they can deny pattern work.
If pattern work is accepted, they should clear you for take-off and issue left/right traffic, They should then ‘sequence’ you (if required) and ‘clear you for the option’.
If they are not accepting pattern work, then they would deny this request (if it was requested at take-off) and after departure, would hand you off to the radar frequency that is active, or if there is no radar facility available, issue a ‘frequency changed approved’ message.
Radar facilities (i.e. approach and departure) are not involved in VFR pattern work. You can of course take-off, check in with approach and request an approach back at the airport, which would be a radar pattern, but if the controller is busy they may ask you to go elsewhere instead (it might be a good idea not to do this on FNF or when you can see we have a lot of inbound aircraft!).
Generally speaking however, all controllers are encouraged to allow pattern work when the weather permits to gain more practice and to offer a better service to the pilots.
Hope this helps and please shout if you have any questions!