The Hima name is associated with various peoples and political entities in the Great Lakes are of Eastern Africa. In recent history the name Hima indicates a sub-group of the Tutsi, originally a Cushite group from the Ethiopian highlands, who entered the area perhaps in the 1300s.

There was a king of Rwanda (a Tutsi Kingdom) named Gahima (1500). [P. Ngologoza, Kigezi and its People, EA Literature Bureau, 1969.]

Some writers think the name Hima is also associated with a Nilotic people who came down from the Sudan along the Nile through Uganda. They conquered the Bantu people in what is now southwest Uganda and the neighboring portion of Zaire.

They were absorbed by the Bantu people and took up their language. Their name in the language is the primary identification of their origin.

One tribe (or social group) of people in Ankole, Uganda, are called Hima (Bahima). The Hima speech is classified in the Ethnologue as a dialect of Nyankore (Nkore/Nkole). The Ethnologue notes, however, that this speech form "may be a separate language."

The relationship of those speaking HIma and other varieties of Nyankore would likely be the same as with the dialects of the Cushite Tutsis and Bantu Hutus speaking Rundi/Rwanda.

There is no listing for the Hima in Tanzania. It appears the Hima are limited fairly much to their original settlement area.

Hima and Tutsi
The general view of the Tutsi is that they have come from a Cushite background in northeastern Africa. Some scholars have seen connections with Nilotes. Over the centuries the similarities in the situations of the Nilotic and Cushite upper class possibly led them to be more closely associated.

In modern times, they appear to be so closely associated that they can be considered one general class across the variety of speech in the various political divisions of the Lakes Region. Their varieties of Bantu speech are very similar.

President Yoweri Museveni of Uganda is a member of the Hima group of Uganda. He has even been the target of charges that he is part of a broad Tutsi-Hima plot to conquer the region. In current discussions of this matter, the Hima are commonly considered a sub–group of the Tutsi. In Uganda, a comparable tribe, or social group, to the Hutu are the Iru (Bairu).

Check this link for more on the Hima-Tutsi connection

Language References
I find no listing for a language or dialect of Hima or Kiga in Tanzania, but Holoholo is related, and stated in the Ethnologue to be also in Zaire, but interestingly enough, not in Rwanda or Uganda!

Nyankore and the speech of the Hima is related to Kiga (also spelled Kyiga, Chiga, pronounced Chiga), a language group of Southern Uganda related to Kinyarwanda. I note in the Ethnologue that one dialect of Rwanda is called Tshiga. This would be the same language group.

The degree of lexical similarity of Nyankore to Kiga is in the range of the Ethnologue’s classification for one language (normally 90-95%). I note in the listing for Chiga (Uganda), however, that mutual intelligibility is shown as only 72%, which would be low.

Speech and Politics
Note that what are classified as "dialects" are often political designations. Thus speakers of the Tshiga dialect are most likely simply members of the same language/ethnic group called Kiga in Uganda who just were on the wrong side of the border when the Germans and English drew it!

This is a problem we must watch for all over Africa, even being aware that when we say, for instance, the "Zambia Tonga" and the "Zimbabwe Tonga," etc., that we are probably indicating political distinctions, not true people group distinctions.

Also related:
Hima, Ham and Cush
Tutsi, Hutu and Hima -- Cultural Background in Rwanda
Tutsis -- the Ethiopia-Somali Connection