So first of all you are confusing tabbing with focus. While a screen reader user may use arrow key navigation, they may also bring up a list of links on the page, or forms, regions, tables or (most often) headings in order to navigate to sections that interest them / get a feel for the page layout.
Tabbing is useful as there are plenty of people who cannot, struggle or prefer not to use a mouse but do not require the assistance of a screen reader i.e. Cerebral Palsy due to accuracy issues.
Hopefully that clears that bit up for you.
As for autofocusing an input on a multi-page form, that is more of a UX issue...is it good UX for all users if you auto-focus the form.
I would say yes, requiring people to click, tab or use shortcuts to get to the input if that is going to be the action that 99% of people need to take is not good UX.
Notice I say the <input> and not the heading.
What you do want to do is look into how to indicate this is a multi-page and multi-step form on the first page of the process.
You should also consider having a checkbox as the first step that says something like "this form is multi-page, would you like us to auto-focus the first question when a new page is loaded" and have it default to checked.
That way someone has the choice to stop the auto focus if they wish.
Additionally (or as an alternative if you do not want the checkbox) you may consider adding some text near to the submit button (and properly associate it with aria-labelledby and an ID) that explains that the next step loads a new page and the first input will be auto focused.
Another thing is that assuming you have titled the pages correctly (Form name or process - step 2 of 5) then a screen reader user can easily ensure they are on the correct page.
So focus the input if this is a multi page form, not the heading level 1.
Focusing headings after navigation is a trick for Single Page Applications (SPAs) so that might be where you have seen that, if you load the page content via AJAX then that is one of the recommended ways to handle navigation. You indicated next page "loads" which makes me think this is a Multi Page Application, but I thought I better add that just in case as a general tip.
Bear in mind that all of this advice is general. Depending on your form, site design, loading pattern, user base (experienced users / internal system vs general public) etc. the best practice may be different.
So long answer short - use autofocus on the input if this is a multi step form and you think it is appropriate and obvious that the input would be the next thing to fill in.
Then test it with someone who uses a screen reader and get feedback as to whether it makes sense or if it was confusing or unexpected.