The adventure introduces Gundren Rockseeker simply as a "dwarf" and doesn't seem to mention what kind of dwarf he is. I've also tried to look for information about his brothers and it describes each those as simply being a "dwarf" as well.

So, in the Lost Mines of Phandelver text, are they ever referred to as Mountain Dwarves or Hill Dwarves?

It's just occurred to me that the name "Rockseeker" might be an established dwarf surname in Forgotten Realms lore, which I'm not all that familiar with, so if someone can provide an official reference to any information on the Rockseekers being Mountain Dwarves or Hill Dwarves, even if it's from other editions of D&D, then that would also be an acceptable answer, but I'd still prefer it if the answer came from the Lost Mines of Phandelver.

^{The reason is because I'm setting this in my home universe rather than the Forgotten Realms. The different dwarves have different accents. Mountain Dwarves have the stereotypical Scottish accent, but Hill Dwarves in my universe have northern accents (that is to say, North of England, specifically Geordie or Yorkshire accents), so this will affect how I play Gundren (since I like to try and do the voices for my characters if I can).}

^{If it turns out that the adventure simply never states what kind of dwarves they are, then I'll simply pick one, but I'd prefer if I play them as the correct kind of dwarves if it does say which kind they are...}

They are (arguably) Hill dwarves

There's nothing to go on in the module book.

You're right, it is never stated explicitly anywhere in the module what kind of dwarf the Rockseeker brothers are. Whenever they are mentioned in the narrative they are just referred to generically as dwarves.

If you ever need stats for them in the game, the module suggests using generic commoner stats (LMOP p.41), so it's not possible to use a stat block to work out which racial characteristics they've been assigned.

But, there is another source you may consider authoritative:

However, LMOP also comes with pre-generated player characters (included in the starter set and freely available online) with suggested backstories that tie them in to the events of the plot.

One of the pre-generated characters is a dwarven cleric, and in the 'Bonds' section of their character sheet it says:

I have three cousins—Gundren, Tharden, and­ Nundro­ Rockseeker who­ are­ my friends and cherished clan members.

This pre-gen character is a Hill Dwarf, so it seems a fairly safe assumption that their 'cousins' and 'cherished clan members' would also be Hill Dwarves.

Not directly specified

I searched the entire adventure text on DNDBeyond, but the adventure never provides more details about the Rockseeker brothers other than that they're dwarves. Gundren and Nundro use the generic commoner statblock if their stats are relevant/necessary.

...but one pregen might provide the answer

The Starter Set comes with a set of 5 pregenerated characters. One of them is a hill dwarf cleric with the following listed under "Bonds":

I have three cousins—Gundren, Tharden, and­ Nundro­ Rockseeker—who ­are­ my friends and cherished clan members.

This would suggest that the Rockseeker brothers are also canonically hill dwarves.

There is a strong argument for Mountain Dwarf

In the description of dwarves in the Sword Coast Adventurer's Guide (p. 103-104), it says that the "Shield Dwarves" from the Sword Coast and the North are Mountain Dwarves, and that the "Gold Dwarves" from the Great Rift are "Hill Dwarves".

Since the adventure is located near Neverwinter, which, by the way is very near to the Dwarven Kingdom of Gauntlgrym (Shield Dwarves), and that there's nothing mentioning the Rockseeker brothers being immigrants, (the Great Rift is very very far away from the Sword Coast), I would say that they should be Shield Dwarves, specially if you want to use it as a narrative tool.

I think the adventure writers didn't put much thought on this issue, and simply picked Hill Dwarf subrace for the pre-gen character because of the +1 bonus to wisdom, so that the character would be a better cleric.