In addition to the earlier answers, may I only point out that saying "Nurse Jane Doe" (with N usually capitalized) or "accountant Thomas P" tends to have the effect of making the profession a sort of title and emphasises the identity of the person, who is often known to both speaker and listener:
I spoke to Nurse Joan and she says that hepatitis is dangerous.
They have arrested accountant Thomas P for assisting in tax fraud
whereas using the "a" before the profession makes the person's identity less central to the situation, and emphasises the profession itself, as in
the news anchor interviewed a nurse Maria Kutty from Kerala who said that immigration issues are still challenging for Indian workers in Europe.
The FBI is questioning an accountant Thomas P for allegedly assisting organized criminals in money laundering.
Here the profession is emphasised and the person's name is additional information (which can even be put in parantheses because it is not integral to the situation).
It is not typical in standard formal English to write the name of the person after "a nurse" or "an accountant" but this may be much more common in news reporting, informal writing or the spoken language. In formal writing, the solution is to write the name with a comma before "a + profession" or write the name after "a + profession + comma":
the news anchor interviewed Maria Kutty, a nurse from Kerala, who said that immigration issues are still challenging for Indian workers in Europe.
the news anchor interviewed a nurse, Maria Kutty from Kerala, who said that immigration issues are still challenging for Indian workers in Europe.
One situation where "a + profession + name" is actually used formally is when announcing or referring a person who is presumed to be not personally known to the listener:
I am referring a nurse Maria Kutty from India for your expert advice on immigration issues.
Sir, a nurse Maria Kutty is waiting to see you.
Note that if she is presumed known to the listener then it would be
Sir, Nurse Maria Kutty is waiting to see you.