Paying tax on wrong tax code? Who is responsible for changing this? - my employer or HMRC?

I recently changed jobs, and there seems to have been some mix up and I now seem to be paying tax on the code BR - I think this is some sort of emergency code/ or one that is used for a 2nd job.

HMRC wrote to me to find out what my current status of employment was and I wrote back to tell them that the job I currently have is full time - and my only job.

I also contacted HR in my new job as my last month's wage I estimated that I had paid too much tax. They said that they would send me a P46 as the problem may have been that they hadn't yet received my P45, which I have already given to my new employer. Anyhow, I still have not received a P46 from them, and according to my calculations, I have still paid too much tax this month.

My salary is 拢13000. How much tax/NI should I be paying on this (per month), and who's responsibility is it to sort it out.

I done my calculations on listentotaxman.com - are these accurate? (I have not yet received my pay slip this month) so I cannot se

*see if my tax code has been updated.

Does anyone know what my actual tax code should be anyway?

Unfortunately you are responsible for sorting out code number problems. Who did you give form P45 to? A manager? Would suggest you follow up, and ask who they gave it to, and so on, up the line. As it hasn't reached HR, I'll bet it is sitting on someone's desk.
You can also contact HMRC. As P45 hasn't reached HR, and form P46 not done, I expect you are still registered at old employer's reference. HR can tell you new tax office reference number and contact details. Ask for your record to be transferred to new reference, and for correct code number to be sent to employer. If two employers are in same area, can be done at once. If in different areas, could take a couple of days.
You can also get photocopy of P45/1a, which you should still hold (don't let original out of your sight), send to HR, and ask them to check code number with HMRC.
Most people in the UK have a code number of 543L this year. If this is right for you, I would expect you to pay tax of 拢126.08, NIC of 拢69.12 per month. www.listentotaxman is right, although it depends what calculator employer uses, some give personal allowance of 拢5439 instead of 拢5435, so could be pennies out.
You don't say when your old job finished, or when new one started. If you suffered BR deductions before 5th April, send form P60 to tax office, together with form P45/1a (original forms needed here, photocopies not acceptable) and ask for computation and refund.