Your tax should work itself out - provided HMRC sends your employer a correct tax code. At that point your employer will deduct less tax in order to 'repay' the tax you've overpaid. You might have to call HMRC if it doesn't happen automatically.
When this happened to me (years ago) I actually got a refund (negative tax deducted) in the month where my employer applied my correct tax code. (but don't forget that this is a one off when the tax code is corrected and you don't get bumper paychecks every month)
As fas as NI, I believe this is calculated based on Sundays you were working. I cannot explain the precise numbers but if you have just started work there may have been only one (or even zero) Sundays in the period of employment so your NI amount might look strange (I don't really understand how NI works - so this explanation might be completely wrong - wait to see what others say.
One complication - if your employment started last year (before April 6th) then there might be some small errors in tax charged and NI charged. This is another area that I don't understand. When I first started work [mumble] decades ago, I had one week's employment in the payroll period at the end of the tax year. But it wasn't possible to add me to the payroll for end March in time. So I was paid that one week as extra in the April payroll. I felt I should have paid no tax for that one week (negligible earnings in the year) but HMRC told me it was taxed correctly in the following tax year. Over the years I've been told that HMRC was correct and HMRC was wrong. We are talking about 50 tax, which seems trivial now, but would have made a big difference then.
So, it may take a month or two for things to make sense and be comparable with what online calculators show. And other than paying for professional help, you might just have to accept the quirks for the beginning of the year and assume HMRC/your employer get it right.