Listen to all voice mails on iPhone continuously
Visual Voicemail—where you can select which message you want to listen to instead of being forced to listen to them in a specific order—doesn't seem to allow continuous playback. Also, is suspect that the delay in retrieving a message is because it's contacting the carrier.

It may be possible to find an app that allows playback as you want, but beware of buying a an older app that may not work with more recent iOS versions or one that won't work with your carrier.

I don't know what version of iOS you're running, but more recent versions (I'm looking at 10.2) have an improved interface for playback as well as transcription of the message.