It's widely known that William Carlos Williams actually did have a grandmother from England. So, the chances are very high that the subject matter of this poem is plucked straight from his own life. Of course, it's never really a good idea to think of the poet as the speaker when you're analyzing, though. Poets tend to get all poetic and embellish whatever they're writing about, so we'll just analyze the speaker based on what the text give us.
On the surface, the poem actually doesn't tell us much at all about the speaker. He narrates the story, but his focus is on his grandmother, not telling us about himself. The speaker actually gives a detailed description of her. By the end of the poem, we know she's "wrinkled and nearly blind" (5), and that though she has dementia, she still has moments of clarity that reveal a sharp wit. Also, we know she's English, because that's like... you know... in the title.
Even though we don't get a direct description of the speaker, we can learn things about him based on what he chooses to share. He kicks the poem off by saying, "There were some dirty plates/ and a glass of milk/ beside her on a small table/ near the rank, disheveled bed—" (1-4). It doesn't exactly paint a pretty picture, right? So we figure this speaker isn't one to shy away from the grim details.
The speaker also makes a point of moments where his grandmother gets all sassy. Since he's bothering to tell us this story, we're going to assume that he admires this about his grandmother. The whole poem builds up to her quippy final words, so we think the speaker is a man who admires sharp wit and people who go against the grain. His implied admiration, though, somehow makes his grandmother's impending death all-the-more bittersweet. Sad times.