I'm just off the phone to customer service and they weren't able to help me, but I've figured it out so figured I should post it here in case anyone has problems.
I had somehow deleted my SMSC number and wasn't able to send texts, but was able to receive them.
I found out the SMSC number +353830000800 but when that's entered on its own it doesn't work through the 4636/info menu.
So I rang customer service and they didn't even know where to enter the SMSC number and couldn't provide any help at all, although it wasn't unsurprising.
So I went back to Google and eventually figured it out.
I'm not sure if this applies to all phones, because some OEM skins might have the option built in to change it in the messaging menu, but for those that don't; here's how it's done.
Thanks to this thread and this website I figured out that the Three SMSC number that needs to be entered isn't +353830000800.
What you need to do is:
Go to the dialer
Go to Phone Information
Go down to SMSC and enter: 0791538303008000
Then hit update and everything should start working.
Not sure if many will have this problem, but that's how to fix it.
Just want to leave this here in case someone else has the problem and customer service can't help.
Or maybe you want to pass this on to customer service so they can help.
Like I said, it applies to the Galaxy Nexus and possibly all Android phones.
Thanks for sharing your resolution on this! I'm going to note that here in case any others have a similar issue, thanks again!
I need to do this on my Galaxy S2! Seems to be a slightly different process, can anyone advise??
edit: should this be *#*#4636#*#* 4696 doesn't work for me??
Edit 2: 079153830300800011000A worked for my galaxy s2
Yeah sorry that's a typo. I have it earlier in the post that it's 4636 (number pad text for info) so I must have just made the mistake when typing the next time.
Sorry about the typo, I can't seem to edit the post now though.