Where can I find the voice messages that I kept on iMessage

But once I've kept them, how do I access them? I used to be able to find them next to photos in the message conversation details, but that option is no longer there with the update?

Probably you mean Audio messages.

Was iCloud enabled for Messages? --> Send photo, video, or audio messages on your iPhone, iPad ...

To keep your audio messages longer, go to Settings > Messages. Scroll down and tap Expire under Audio Messages, then select After 2 Minutes or Never.

Watch the GIF

This GIF Has nothing to do with it, its just display;laying the option to manage messages stored in iCloud feature. You just scroll to where the voice messages was sent in the message thread there isn't any one unified location to find them you just need to go back in the conversation to where they was sent.