Not the best answer, but I have sense 4 on my HTC EVO 4g LTE from Sprint and I have my work email set up as push in the HTC email app. Sense 5 should be similar on the One. I usually get emails in my work computer's Outlook program and on the phone in the HTC app about the same time. It may depend on where you are trying to get email from and their servers capabilities.
I use Emoze free email app for Android , it provides push to any type of email ,obviously Exchange ,IMAP,Gmail Hotmail but even POP3 which is very limited email protocol.
I've done benchmarks with the stock ICS email client and with several third party clients and I find that Emoze has the advantage in almost every aspect.
This is from another forum but sounds like it may fix your problem - I couldn't get it to push emails either
I talked with a HTC rep. Follow the steps. 1)Remove your current account. 2)Add account. 3)Choose "Microsoft Exchange ActiveSync". 4)Type your e-mail address and password. 5)Don't click NEXT, click on MANUAL SETUP. 6)Type your e-mail, password if empty. 7)Server address: m.hotmail.com 8)Domain name: LEAVE IT BLANK. 9)Username: your e-mail (ex: abc123@hotmail.com) 10)Check! 'This server requires an encrypted SSL connection. 11)Say NEXT, give a name to your account and finish setup. 12)Open phone settings. 13)Choose Accounts&sync. 14) Auto sync should be ON! 15)Open your mail app. 16)Click three dots to access mail settings. 17)Choose Sync, Send & Receive. 18)Peak time sync: PUSH MAIL. 19)Off-peal sync: PUSH MAIL. 20)Now go back and choose Notifications settings. 21)Check! E-mail notifications. 22)Re-boot the device. 23)Send a test e-mail to yourself. 24)Well-done
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