Assumptions.
1. Gorlice-Tarnow ends on schedule in mid September 1915, leaving Russia badly smashed up but still alive.
2. Serbia is crushed October to early December 1915, per the historical attack.
3. Italy and Romania remain neutral.
4. The British military resource shift of June 1915 mostly effects munitions shipments to Russia. This leaves the 1916 Russian army very weak, but the British army in France is at more or less historical strength.
5. Gen Falkenhayn wants an early 1916 west front offensive.
6. Verdun currently lacks a main supply route, as they did historically.
7. The German attack will begin on schedule in February 1916.
8. As soon as the German attack begins France will commence building the Nettancourt-Dugny standard gauge rail line to Verdun. It will be completed in June 1916, as it was historically. Until this rail line is completed the French army will be fighting at a severe disadvantage. This gives Germany 4 months to win a decisive victory. If that victory is not achieved then the French will commence serious counter attacks from Verdun.
9. In LettersTime an additional 18 German high quality infantry divisions will be available, transferred from the East. Their place is taken by the A-H divisions historically used to attack Italy.
10. A-H will loan 20 batteries of 30.5cm siege howitzers (40 weapons) to Germany for the attack. These are the weapons historically committed to the Italian attack. Since most of the Russian border forts were captured in 1915, A-H has little use for these weapons in the east.
A-H units historically committed to Spring 1916 attack on Italy.
18 x high quality infantry divisions.
Most of the A-H heavy artillery and ammunition reserve.
20 batteries of 30.5cm siege howitzers.
German units historically committed to Spring 1916 Verdun attack.
19 x Infantry divisions (13 left bank, 6 right bank).
(Like France, the Heer rotated units through Verdun. But this is the most participating in the attack at 1 time).
12 x 42cm siege howitzers (The entire German supply of purpose built siege howitzers.)
Virtually unlimited ammunition. The Haber process plants are in full operation now.
LettersTime German Order Of Battle for attacking Verdun.
37 x Infantry divisions (twice the historical quantity).
52 x Siege howitzers (4 times the historical quantity).
Analysis.
http://www.history.army.mil/books/wwii/lorraine/photos/maps/MapII.jpg
(Lorraine 1:500,000 topographical map)
With this much siege artillery there is a real chance that Germany can capture the entire Verdun fortress complex within 4 months. Germany has ample infantry to occupy the newly captured ground. As in previous siege operations the German casualties will be light as the heavy artillery will do most of the work.
The siege artillery will be mostly worn out and most of the ammunition used up by this fight. That means Germany cannot conduct another major siege operation for several months. Fortunately there are no more fortress complexes between Verdun and Paris.
This will result in a massive fight between the French and German field armies on the west bank of the Meuse. Both sides will have good supply lines (France has the newly constructed rail line from Bar le Duc. Germany has the quad track rail line from Metz that crosses the Meuse at Verdun.). The French army is still seriously outgunned in mid 1916, so the Heer can win this fight with any luck at all.
As happened historically, the BEF will launch an attack on the Somme in July 1916 to relieve pressure on France. We know how that fight ends.
Thanks to the cutback in British munitions shipments (especially the 400 x 4.5" QF howitzers and ammunition), Russia cannot launch a serious offensive operation in 1916. The eastern front will remain relatively quiet (except for a bunch of whining from Ober-Ost).
If France loses Verdun and the fight on the Meuse west bank then I think they will quit the war, provided Germany offers anything close to reasonable peace terms.

