If you take orders from the order book, then you can check its position on the blockchain under the 'YOUR TRANSACTIONS' tab. What you'll see here is the timestamp information if it's gone through or not.
If you'd like to see it's position on the blockchain, you can click the unique serial number under the 'DETAIL' heading in the same box. This will send you to EtherScan where you can view the live feed for the smart contract on the blockchain.
If you'd like to see if your order has been filled you can check its fulfillment under the 'YOUR ORDERS' tab. You won't see the same timestamp information like you see under the transactions tab because it's essentially sitting on the blockchain waiting for another person to enter and sign off on the contract and execute the transaction.
In the yellow box, you'll see is if the order has been fulfilled or not.
The time it takes for a transaction to process depends on the activity of the Ethereum network. You can check the estimated block clearing time 'Est. Gas Cost' tab which is highlighted in blue on the top right hand of your screen. Under the 'Avg. Block Time' you can see the current estimated time it takes for the past 15 blocks to clear. This will give you an indication for how long your transaction will take on the network, if you have enough gas and a high enough gas price to incentivize miners to clear it faster.
To learn more about gas prices and gas limits (the nuts and bolts of processing transactions), check out our "What are gas fees?" guide.