This list needs to allow the user to jump from one letter to another (say, from A's to Q's). But we're not paginating, but loading everything onto a single page.
Not saying it's the best way, to display an infinitely scrolling, alphabetized list - in fact, there may be no 'best' way, to display this kind of list. I do however, want to point to to an example of where this happens, in (a previous version of) Snapchat.
I know that you're not making a native mobile app, rather a responsive web app, but I still think that you could apply the same design pattern.
It's something to consider, but a few things I like about this pattern, are that the letters (links), mimic the actual alphabetized list in a top to bottom layout. I also like that it highlights the letters as you move between sections.
Like I say, I'm just posting this as a suggestion, and something that may help you to find the solution that best suits your situation.
I guess ideally you will want one solution that works on desktop and another which utilises the mobile OS default approach.
However if you are looking for one solution which works the same on all devices the I would use a dynamic horizontal A - Z, where the letters are links based on the letters used in your dataset, i.e. Depending on your data you probably don't need to show all 26 letters in the alphabet, e.g.
Obviously this could become a problem on a mobile device due to the limited width, especially if you are using most letters in the alphabet, but you could allow them to wrap on to two lines if needed, mainly because you need to ensure the tap targets for each letter are large enough on a smaller viewport.
Imagine the following test data (https://www.mockaroo.com):
Checking the first letter we have the following number of entries starting with:
The most intuitive (in my opinion) would be to present a following slider:
A B E J L M S .
where the user can smoothly slide on. If the page can accommodate only 10 entries (out of 30) I'd start with the one at position
30 x [percent value of the slider] and have the sliding bar (underscores instead of min) covering shown entries (Jimmy,Hayes would be shown first in the page, Linda,Johnston last).