What are the options for booking short-stay accommodation between London and Newcastle at very short notice?

Campsites for people who are walking mostly allow walk-on's, reserving some or even all their lots for people who walk. Most restrict staying to one night, which will suit you.
And when carying your own tent and gear, you can also ask at farms and such for a place in their field or back-yard if you can not find a proper site.

Hostels, both the YHA and the independent hostels allow walk-in stays, they also allow you to phone a few hours before arriving and asking to hold a bed.
Several off the rural hostels I used in England did have one or a few spots for tents, so if they do run out of beds they might still accommodate you.
Whether they will hold the bed or tent spot without back-up (like a credit card) will depend on whether they are busy and whether they have had bad experiences in the past.
And some hostels still have a closed time in the afternoon. In which case you may not reach them by phone.
Some rare ones still use cards, fill out name, drop in the box for them and you are secured of a bed, when out of cards they are out of beds. This only works for walk-in customers, of course.

Hotel booking sites, which also cover B&B's and some hostels also accept bookings on the day, sometimes even in the evening of an evening stay.
Apps for booking sites are getting common, allowing you to make a booking as long as you have data access. Or just checking out availability before walking in.
As per comment by @Calchas:

HotelTonight is a useful app to have on the phone—it shows you the hotels near you and their prices for a room tonight, and enables you to book a room directly from the app—although perhaps hotels are not quite what you are looking for.

While getting to be more uncommon, walking in at hotels is also possible.
On quiet days many will have rooms available, some may only do pre-booked but most will be happy to help you out.

Where available, tourist information offices, and their websites, can be of use, more so in some places than others, but worth trying if you are near.

When out of beds, some hotels and some hostels will help you find a place to stay near, in B&B kind of accommodation, in an other hotel or hostel or even with friends or relatives with a spare room or a spacy back yard where you can pitch your tent.
So if desparate, just try one even when it says 'full'.

Whether it will be successful depends on when you travel and how many others will travel at the same time.
Start looking before you actually need to stop for the night, whether online or in person. And check a couple of days ahead when you have (free) internet, to see whether all hotels/hostel/campsites in an area are booked solid.
In the worst cases you may elect to stay in your last location for one or two more days or use an alternative route or in the worst cases use transport to skip the busy area.

If you follow a long distance path, or several, you will find quite a few places to stay that are used to people walking in looking for a place to stay. Mostly this works very well but on long weekends and during all school holidays these places tend to have more people coming in than they have space for.
Most long distance paths have books and/or websites mentioning accommodation near or on the route, often with warnings when there are not enough beds for popular times.

In some more remote locations there are huts on or near the long distance hiking routes, and those will be mentioned in the route documentation. Mostly they are free to use as long as you bring your own firewood, (or do without a fire, which is fine in summer.)

And talk with the people you meet. Specially those that regularly walk the route or part of it and those going the other way round, for tips and suggestions you will not find in print.

Most of this I have learned when traveling in England (and the rest of the world) by public transport and by bike, I do not use tents but friends do and prefer the smaller campsites where people walk on with all their things on their back.
While at the start of your journey you will not be able to judge how far you can walk, you will work out your average length of route and will be able to book ahead for busy days, like weekend and national holidays.

My personal worst was being between Luton and Stanstead airports while the volcano on Iceland stopped all air traffic. All hotels and motels miles around filled up, but even then I found one where the motel staff told me I surely would.