For a new AD&D campaign, assume the following: all material in the 1st Edition PHB (available to players) is unchanged as written, with one exception below. All material in the DMG (theoretically DM-only) is subject to modification by the DM, as follows.
Core Mechanic
In this fix-up, the AD&D core mechanic can be expressed as follows: roll d20 + modifiers, and try to get a total of 20 or more. This will be seen in more detail below.
Time and Distance
This is the one alteration to material in the PHB (p. 39). The round is considered to be 30 seconds. Indoors, 1" equals 5 feet. This allows miniature figures to be used at an equivalent ground scale; all distances expressed in inches (movement, missile fire, spell ranges, etc.) are unchanged.
Surprise and Initiative
Surprise is as written in the DMG. Roll 1d6 for each party, with a roll of 1 or 2 indicating units of surprise, modified by individual Dexterity. Spells can only be used with 1 or 2 segment casting time, as appropriate (1st or 2nd level).
Initiative is simply a 1d6 roll for each side; high roll goes first. Spellcasters must declare spells prior to this roll, and any successful attack before their action ruins the spell. Ties are broken by low casting time or weapon speed (a difference of 5 or more indicating 2 attacks).
Combat Tables
Combat is performed by rolling d20 + AB + AC + modifiers; if the total is 20 or more, a hit is scored. (A natural roll of 20 is counted as 25.)
AB (attack bonus) is equal to fighter level or monster hit dice, or about half level for non-fighters. As an option, you may use Weapon AC adjustments when fighting NPCs in armor.
Saving Throws
In general, saving throws for fighter-types and monsters are made by rolling d20 + level + modifiers; if the total is 20 or more, the save succeeds. They add +1 vs. rod/staff/wand, +2 versus petrification/polymorph, and +3 for poison/death magic.
Non-intelligent monsters use half hit dice except for poison/death magic. Other classes can roll d20 + about half level + modifiers and then consult the first line of the appropriate DMG matrix.
Ability Checks
Ability checks for "skill"-type activities are performed by rolling d20 + half the applicable ability score + level (if the ability is a prime attribute for the class); if the total is 20 or more, the check succeeds. A prime attribute is any ability that leads to an XP bonus; or, if the class gets no XP bonus, any ability with a minimum score requirement. Exceptional strength adds +1 per category.
Extremely easy situations can give a bonus of +5 to this roll. Unusually difficult situations can give a penalty as large as -5 or -10 or more to a roll like this.
Unearthed Arcana
Unearthed Arcana must be taken as a collection of optional rules. Nonlethal and weaponless combat uses the rules therein. New equipment, spells, and magic items may be discovered in the course of campaign play. New race and class options (including weapon specialization and multiclass possibilities) are considered for NPCs only, unless special DM dispensation is given. Cavaliers and paladins are still sub-classes of fighter (not each other).