one or [the] other in American and British English

The version without the article seems to be preferred in British English while being strongly disfavored in American English. The version with the article is common in both varieties of English.

Having said that, depending on the precise context, the two versions may mean different things.

If the intended meaning is that you should ask one out of more than two grandparents, then there should be no the. However, I agree with user Janus Bahs Jacquet's comment, below, that in American English, one or another would be much more likely in that case.

If there are just two grandparents you could ask, then, in principle, either is acceptable. However, dropping the article seems to actually be preferred in British English, while in American English retaining the article is strongly preferred, and dropping it may sound outdated.

Discussion

General dictionaries

Merriam-Webster only has an entry for one or the other, which it defines as one of two persons or things but not the other (here).

The Oxford English Dictionary (OED) mentions both phrases in its entry for one. So we find the following:

9.a. Used in antithesis to another, other, others. one and another: more than one, two or more in succession. one or other of: one of.

A sample of usage:

The metallic ores are usually embedded in one or other of these minerals.

The OED also mentions an obsolete usage of one or other, with the meaning altogether; one way or another; in all respects. Sample usage (from 1796): Indiana has one or other the prettiest face I ever saw.

Finally, the OED notes that one is used as short for ‘one or the other’, with a remark US Regional. A sample sentence: The driver had to stop or run me over, one.

None of the sources I consulted explicitly mention regional variation, though of course that doesn't necessarily mean there isn't one. For what it's worth, Garner's Modern American Usage doesn't have an entry for either; it does use one or the other twice in the text.

Corpora of English

All things being equal, a larger corpus will inevitably have more hits. Therefore what I will be giving is the ratio of the number of hits to the total number of words in the corpus, a quantity I will call the scaled number of hits.

    Corpus of Contemporary American English (COCA)
        scaled numbers of hits (×10 ^{- 7}):
                one or other           1
                one or another      9
                one or the other   18

    British National Corpus (BNC)
        scaled numbers of hits (×10 ^{- 7}):
                one or other        39
                one or another      6
                one or the other   15

(The raw numbers of hits were, respectively, 60, 505, and 1034 in COCA, which has a total of 560 million words, and 390, 64, and 150 in BNC, which has a total of 100 million words.)

These results suggest that, in American English, one or other is largely disfavored: one or another is 9 times more frequent, and one or the other 18 times. In contrast, in British English, one or other is actually preferred, being 6 times more frequent than one or another and 2.6 times more frequent than one or the other.

Comprehensive grammars

Both CGEL and ComGEL freely use both one or other and one or the other, and they use the former even when there are just two alternatives. CGEL in particular uses one or other a lot, both in the body of the text and in the exaples. On the other hand, Longman doesn't use one or other at all, and only has three occurences of one or the other.

It should be noted that CGEL mentions (p. 1330) that one or other can also be used as a fixed phrase, which functions as a determiner meaning "either/any".

Examples of usage from google books

As far as one or other, consider the following examples of usage:

These different forms which warfare takes according as one or other of the three arms preponderates… (source)

I have also been very successful with solutions of ferric salts which are precipitated by ammonia either alone or with one or other of the above salts and acids. (source)

In these cases, it is clear that other refers to elements of a set that has more than two members.

However, there are also examples like this, where one or other is clearly used to refer to members of a set of just two things:

… two separate windings arranged within said hollow core and surrounding the said cores, two connected movable iron plates arranged at the opposite ends of said case and core, a holder for said case, two insulated fixed switch contacts connected each to one end of one or other of said windings… (source)

From searching through google books, however, one gets the impression that one or other was more likely to be used in times past. Google Ngram is consistent with that impression: