They say that there's no rest for the wicked, don't they? ;-)
For YouTube:
Start from the box's home page. (i.e. press the house symbol at the top right of the remote control).
Press the right arrow and, if necessary. the down arrow until YouTube is highlighted.
Press OK.
After a bit of a wait you'll get to a screen with a red menu list on the left. If you've got an account (and want to watch your stored videos) press the up button to get to 'Sign In'. Otherwise press up to get to 'Search'. Press OK.
To search: Use the arrow keys to select the first letter of your search term. Press OK. Repeat as necessary. Then move the cursor down to an illustrated suggestion below or up to the written ones underneath search field and select OK.
For radio stations:
If you're simply after Radio 2 or similar forget about using your Now TV box and simply access them via Freeview (or Freesat or whatever else you've got). The Freview channel numbers are in the 700s. (e.g. 702 for Radio 2, 704 for Radio 4, 708 for Radio 4 Extra, 715 for Magic, etc)
For stations from around the world, start from your Now TV box's home page and navigate across to the Tune In app. Press OK. Press the left arrow to get to Search and press OK twice (Alternatively, if you've been there before, stations you've listened to will be under 'Favorites' - horrid US spelling!). Use the arrow keys, plus OK, to search for stations.
Hint: Some stations have servers that are slow to respond and, after pressing 'Play' you simply see the word 'Buffering'. If that happens, press the 'Up' arrow, which takes you back to the station name. Press 'OK'. You'll then see 'Pause'. Press 'OK'. You'll then see 'Resume'. Press OK and you should be able to hear the station.