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My father was a CUKC in 1974 and as per law he became a British citizen on 1 January 1983. My mother could never come to UK despite my father's initiative to bring her to UK. My mother again did not submit application form for British passport. In fact, my mother wanted my father to be in Bangladesh which is our home country as my father was suffering from number of diseases. However, my father left for UK in late 1980 and died in 1981. Since then any attempt to come to UK by any of our family members have never been undertaken. My father being burried at UK, we feel a need to come to UK. At this situation, I like to know, to ease visit to UK, if my mother has been registered as British citizen. If she was, which is the place where I can get this information? It is mentionable, my mother is not able to say anything regarding this.

The route to British citizenship for spouse of a British citizen is through "naturalisation", and one cannot qualify for naturalisation without having lived in the UK.
So, if your mother has never entered UK till date, then her being a British citizen isn't a possibility.

rajaiizahir wrote:My father was a CUKC in 1974 and as per law he became a British citizen on 1 January 1983.

But, as per you, your father wasn't alive in 1983:

rajaiizahir wrote:However, my father left for UK in late 1980 and died in 1981.

If she was married to him, then she may apply for a Certificate of Entitlement to the ROA.

Yes he died in 1981 yet as per law he would have become a citizen had he been alive. Although this carries no value to the deceased person but certainly carries values to his heirs.

By the by, can anyone tell if a woman remains entitled to ROA if she after death of the husband gets married to another person and claims ROA during the subsistence of the second marriage?

In response to pennylessinindia I just cut and paste from the UKBA website:
[If you were issued with a British passport between 1 January 1973 and 1 January 1983 it may be stamped with “citizen of the UK & Colonies” and “the holder has the right of abode in the UK” or “the holder has patriality in the UK”. These stamps would indicate that the holder became, or if deceased, would have become a British citizen on 1 January 1983.]

You will get it in 'Confirmation of Nationality Status' page. This means irrespective of the fact if someone was alive or dead being a CUKC he would be entitled to become a British citizen on 1 January 1983. This rule is just to offer nationality benefit to the heirs of the deceased who was a CUKC.