Given that Apple slows down your phones to save battery, how do I speed my iPhone 5 back up now I have replaced the battery myself?
The iPhone 5 is not affected by the slowdown, only the 6 & higher - see How can I verify if Apple is slowing down my phone because of an old battery?
In addition to Tetsujin's answer, even if your phone were affected, if you've replaced the battery, then your phone would be working at full speed again. The throttling is based on the phone's internal diagnostics of the battery condition. Assuming the new battery you put in is equal to or better than the original battery, then the phone would automatically un-throttle.