I believe my employer has an incorrect NI Number/overpaid income tax

TheCant

I'm 23. I received my national insurance number years ago and saved the number on my phone so I wouldn't forget it.
I'm an university student and got a summer job in July 2018 - I work mainly summers but sometimes during half terms (it's in a boarding school).
I was unaware there was such a thing as income allowance, I found out just last friday and I believe I've been paying income tax since I was first employed. 
The most recent payslip I received states my tax code is 0T M1, I checked the tax code meaning it according to gov.uk it the 0T means:
"Your Personal Allowance has been used up, or you’ve started a new job and your employer does not have the details they need to give you a tax code"
The M1 means an emergency tax code, however, in another page it states only these tax codes: 1250 W1, 1250 M1 and 1250 X are emergency codes, so I'm not sure what to make of it.
However, I believe the biggest issue is that I think my employer has an incorrect NI number (one digit is wrong). As I said I save my NI number on my phone so I'm 90% sure the number I have is correct.
Due to the employer having an incorrect NI number I think my tax code is wrong as a result.
What I tried to do so far:
Create and log into a personal income account - I created the account but when I try to log in it says there is a problem: "Your sign in details have not been recognised by HMRC" I contacted HMRC by email (phone service was closed friday night) and they said the that means the details I have provided to the verify agency do not match the details HMRC holds for me.
I tried to confirm my NI number through web chat - They said they couldn't find me on the system.
I filled a form to verify my NI number and sent it by post just now - Apparently it can take 15 days to confirm.

I have a few questions:
Once I confirm my employer has an incorrect NI number for me, how do I go on about changing it? Should I contact HMRC, my employer or both?
Once the NI number my employer has is corrected, would I be able to get an income tax refund easily?

Marcon

TheCant said:

I have a few questions:
Once I confirm my employer has an incorrect NI number for me, how do I go on about changing it? Should I contact HMRC, my employer or both?
Once the NI number my employer has is corrected, would I be able to get an income tax refund easily?

Both.

In theory yes, but don't bank it being simple and sorted out in a few days. If you've allowed the use of incorrect information to run for some time, it's not unreasonable for HMRC to take a bit of time to sort it out. With luck, though...

Comms69

Learn your NI number, its 9 characters. It's literally the single number that identifies you to a huge part of your life - employment, pension, benefits etc.

TBagpuss

If HMRC are saying that the numbers don't match up you need to be clear as to whether you may have the wrong number - do you still have the NI card you were sent when you turned 16, to double check that the number on your phone is correct? 
How did your employers get the number they have? Did you give it to them? 

Savvy_Sue

TBagpuss said:

If HMRC are saying that the numbers don't match up you need to be clear as to whether you may have the wrong number - do you still have the NI card you were sent when you turned 16, to double check that the number on your phone is correct? 
How did your employers get the number they have? Did you give it to them? 

I think that at least some payroll systems will let the employer check that they have used a valid NINo (I intend to check this next time I'm doing stuff on ours ...) But if that check is not carried out, then the wrong NINo can be used. 

So is the NINo on your payslip what you think it should be?