Both depend on the spreading factor (SF) you use. If you have good reception and are at SF7, you can transmit your message in tens of milliseconds to your gateway. If you have bad reception and are at SF12, it can take up to 1,2 seconds. At SF7 you will be able to send more messages because each message takes up less airtime. If you keep the messages short, you can also send more. See this thread for airtime calculation.
The transit time from the gateway to your application completely depends on the solution you choose. On TTN and with a gateway connected through wired Ethernet this will only take some tens of milliseconds (at current load levels). If the gateway uses a slow cellular connection this delay will of course add up. If suddenly tens of thousands of people start using TTN, it will probably slow down too.
Remember that the TTN service is ‘best effort’. In a commercial setting you will probably sign (and pay) SLA’s with commercial providers which will indicate the maximum latency between gateway and your application.
I agree with the wording of @Gig that LoRaWAN is not for real time communication but for (short) message sending. Like you would send an e-mail or a text message. Latency isn’t the factor determining the quality of the service here, as long as the message gets delivered in a timely fashion.