Replace door knob with lever handle on rim latch door
If the latch is in good condition, you could reconsider just replacing the knobs with new lever handles.  Yes, they'll be "inboard" (towards the center of the door), but you'll maintain the hardware.

Replacing the latch hardware

New rim latches are available, but they're rare in the United States.

You'll typically replace the latch assembly (on the door) as well as the striker (on the jamb) to ensure you have alignment between the two.   

You need a mechanism which is furnished with a square hole, as well as a lever set with a matching square spindle (shaft).

It may be more typical to source these from Europe.  Here is one example of one that's currently available from a global hardware manufacturer:


  
  Union latch model 1209 from Assa Abloy


This is designed to be reversible, for either right-inswing or left-inswing doors.  The upper striker appears to be a "privacy lock", for example for a bathroom or bedroom -- the British may call this a "snib lock".*

Then using a lever set with square shaft, you're all set.  A "set screw" will hold the levers in place.  (Apparently in Britain a "set screw" is known as a "grub screw"!)   One lever mates to the latch box directly.  The other uses a small trim plate (escutcheon), as you'll see in the photo below.

Typically European latches use a 7.6 mm spindle and 8 mm hole (source), and antique hardware may use a 7 mm spindle.  

European suppliers are used to furnishing matching compatible handle sets and can provide the correct length of square spindle as required.

Here are some current sources (2019) that you might use for starters:


  
  eBay seller 


eBay UK, Doorhandleparts.com, Locks Online Limited, Timothy Wood Limited, Duffels Locks, and find tons of others by searching "Buy Union 1209 latch" or "Buy rim latch UK".


  *⚠️ Caution about the privacy lock: In the USA, local laws or building codes may require a way to override this lock, for example using a small hairpin, so you don't have to kick the door in if the person inside becomes incapacitated or can't open the lock by themselves.  Verify that the lock you buy has this feature, or get a version without the snib-lock if needed.


A word about "backset"

Your question asked for a rim latch "with the spindle hole near the edge of the door".  This dimension is called the backset.

On a modern, tubular lock, there are usually different choices of backset depending on the door, and hardware that's sold "off the shelf" is often built to be adaptable so the store doesn't have to carry multiple sizes.

Backset dimensions of tubular locks, and the Union rim latch, are shown here:


    


It looks like the latch in the original photo has a backset of around 6—8". If you replace it with a new rim latch, you could obtain a latch with much smaller backset. Here are some examples:


Rim latch with 1-1/2" Backset, House of Antique Hardware
Rim latch with 3-1/16" Backset, Amazon.com
The Union latch model above has a 76 mm (~3.0") backset.  Union Online


I don't think you can go much smaller than 1-1/2", since this position sets up how close to the doorjamb the knob, and your hand, have to go!