I finally managed to defeat Almalexia today, but when I exited the Mournhold Temple, I found that Azura wasn't waiting for me. Gaenor was outside and ready to attack, but would that cause Azura to fail to show up? Here is my load-list if it helps, but I'm not using any mods that affect the Tribunal MQ:

GameFile0=Aduls_Master_Mod.esm
GameFile1=Bloodmoon.esm
GameFile2=Havish.esm
GameFile3=Hilgya the Seamstress.esm
GameFile4=Morrowind Patch v1.6.5-BETA.esm
GameFile5=Morrowind.esm
GameFile6=Siege at Firemoth Fort.esm
GameFile7=Texture Fix - Bloodmoon 1.1.esm
GameFile8=The Undead.esm
GameFile9=Tribunal.esm
GameFile10=TR_Data.esm
GameFile11=TR_Map1.esm
GameFile12=TR_Map2.esm
GameFile13=abotWhereAreAllBirdsGoing.esp
GameFile14=adamantiumarmor.esp
GameFile15=Aduls_Arcane_Armature.esp
GameFile16=Aduls_Leggings_Bloodmoon.esp
GameFile17=Aduls_Leggings_Tribunal.esp
GameFile18=Alakazar's Lamp_no_music.esp
GameFile19=almalexia armor.esp
GameFile20=Alternate Beginnings 2.esp
GameFile21=Amulet of Scrye.esp
GameFile22=Animated Morrowind 1.0.esp
GameFile23=Antares' Creatures - Cecaelia.esp
GameFile24=Apostates of Oblivion_2.1.esp
GameFile25=Appropriate Greetings M - T - 1.5.2.esp
GameFile26=AreaEffectArrows.esp
GameFile27=AscendedSleeperSFX.esp
GameFile28=Ashlandic Marauder Armor.ESP
GameFile29=Atronach Expansion X.ESP
GameFile30=Balance - items.esp
GameFile31=Balance - Magic Effects.esp
GameFile32=Balance - NPC Spellcasting.esp
GameFile33=Balance - spells.esp
GameFile34=Barabus' fireplaces 2.esp
GameFile35=BB Dark Brotherhood by Westly.esp
GameFile36=bcsounds.esp
GameFile37=Better Bodies.esp
GameFile38=Better Clothes_v1.1.esp
GameFile39=Better Propylons MI.ESP
GameFile40=Better Skulls.ESP
GameFile41=BetterClothesForTB.esp
GameFile42=BH_DarkElves.esp
GameFile43=BH_Helseth_Replace.esp
GameFile44=BH_Imperials_M.esp
GameFile45=BH_Redguards_M.esp
GameFile46=BH_WoodElves.esp
GameFile47=bones.esp
GameFile48=Books of Vvardenfell_2.0.esp
GameFile49=Capital Guard Addon.ESP
GameFile50=Caves of the bitter coast v 1.1.ESP
GameFile51=CB-illusionbolt1.0.esp
GameFile52=Chalk30-Base.esp
GameFile53=Clean Aduls_Arsenal.esp
GameFile54=Clean Aduls_Leggings.esp
GameFile55=Clean Aln'Baldrahn.esp
GameFile56=Clean BadKarma Clothing Vendor.esp
GameFile57=Clean Better Daedric.esp
GameFile58=Clean chapelsv2.esp
GameFile59=Clean Deepest Chasm1.1.esp
GameFile60=Clean Lights_300_V4.esp
GameFile61=Clean Welcome to the Arena!.ESP
GameFile62=Clean_Templar_Pauldron_Fix_v11.esp
GameFile63=Creatures Great House Dagoth Patch.esp
GameFile64=Creatures.esp
GameFile65=Dagger of Symmachus.esp
GameFile66=DN_AshVampires.esp
GameFile67=Dracus's Durzog Armor Tribumal V1.1.esp
GameFile68=Dwarven creatures.esp
GameFile69=Dwarven Spectre Additions.esp
GameFile70=EBQ_Artifact.esp
GameFile71=EmmasKhuul.esp
GameFile72=entertainers.esp
GameFile73=Exotic Imports by Westly.esp
GameFile74=Expanded Sounds.esp
GameFile75=farrp_Quivers-FletchingKit_V2-5.esp
GameFile76=Fligg's Slave Mod.esp
GameFile77=Frost_atronach_solsthiem.esp
GameFile78=Gameplay - Dialogue.esp
GameFile79=Glowing_Mines.esp
GameFile80=Go To Jail 3.5 - T B.esp
GameFile81=Golden Saint Armor.esp
GameFile82=Great House Dagoth ( Master Index Compatibility ).esp
GameFile83=Great House Dagoth.esp
GameFile84=Greater Undead.esp
GameFile85=Haunted Tombs v1.06.esp
GameFile86=Havish Mini Patch.esp
GameFile87=Healers.esp
GameFile88=Hilgya the Seamstress Expansion 1.esp
GameFile89=IceNioLivRobeReplacerALL.esp
GameFile90=Ice_ElegantGowns.esp
GameFile91=Illuminated Order v1.0 - Bloodmoon Compatibility Extras.esp
GameFile92=Illuminated Order v1.0.esp
GameFile93=Imperial Chain Boots n Gauntlets.ESP
GameFile94=Key Replacer Trib & BM.esp
GameFile95=Kn-PowerToThePeople.esp
GameFile96=K_BugShield_v1.0.esp
GameFile97=LeFemmArmor.esp
GameFile98=Less_Generic_Bloodmoon.esp
GameFile99=Less_Generic_Nerevarine.esp
GameFile100=Less_Generic_Tribunal.esp
GameFile101=LGNPC_Aldruhn_v1_13.esp
GameFile102=LGNPC_AldVelothi_v1_20.esp
GameFile103=LGNPC_GnaarMok_v1_10.esp
GameFile104=LGNPC_HlaOad_v1_32.esp
GameFile105=LGNPC_Indarys_Manor_v1_45.esp
GameFile106=LGNPC_Khuul_v2_01.esp
GameFile107=LGNPC_Khuul_v2_10.esp
GameFile108=LGNPC_MaarGan_v1_11.esp
GameFile109=LGNPC_NoLore_v0_83.esp
GameFile110=LGNPC_PaxRedoran_v1_12.esp
GameFile111=LGNPC_Pelagiad_v1_13.esp
GameFile112=LGNPC_Pelagiad_v1_21.esp
GameFile113=LGNPC_Secret_Masters_v1_21.esp
GameFile114=Lgnpc_SN.esp
GameFile115=LGNPC_SoulSicknessPatch_v1_00.esp
GameFile116=LGNPC_TelMora_v1_11.esp
GameFile117=LGNPC_TelUvirith_v1_10.esp
GameFile118=LGNPC_VivecFQ_v2_03.esp
GameFile119=LGNPC_VivecRedoran_v1_42.esp
GameFile120=Longspears.esp
GameFile121=MagickaRegen-v20.esp
GameFile122=Mashed Lists.esp
GameFile123=Master Thief's Privileges 1.2.esp
GameFile124=master_index.esp
GameFile125=MDP - Fort Buckmoth.ESP
GameFile126=MergedMods.esp
GameFile127=Minions of House Dagoth - No Undead.esp
GameFile128=Misc_Tweaks.esp
GameFile129=Nasty Camonna Tong.esp
GameFile130=New Argonian Bodies - Clean.esp
GameFile131=New Khajiit Bodies - Clean.esp
GameFile132=Nudity Greeting Expansion V1.esp
GameFile133=NV_BB_Imp_female_Armour.esp
GameFile134=NX9_liches.esp
GameFile135=Of Love and Ignorance.esp
GameFile136=P.R.E. v4.0.esp
GameFile137=pcc_dunzar_02.esp
GameFile138=PCC_Knaarus_02.esp
GameFile139=pcc_sotha_sound.esp
GameFile140=Pearls Enhanced - Colored Pearls Add-on.ESP
GameFile141=Poe Lighthouse.esp
GameFile142=Redaynia Village.esp
GameFile143=ReptileInseparableSiblings.esp
GameFile144=Scissors & Thimbles III.esp
GameFile145=Scissors & Thimbles.esp
GameFile146=Scrolls of Khartum.esp
GameFile147=Sea_of_Solstheim.esp
GameFile148=Silt Strider Armor - Ashlander Guards.esp
GameFile149=Sixth House Advanced 2.1.esp
GameFile150=sm_creatures_02.esp
GameFile151=sm_DwemerStaves.esp
GameFile152=Sol-Weather Tweak.esp
GameFile153=Solstheim Sauna at Thirsk.esp
GameFile154=South Wall.ESP
GameFile155=Telos Rin Graveyard.esp
GameFile156=The Neverhalls.esp
GameFile157=The Runaway ver 1 2.esp
GameFile158=The Sable Dragon.esp
GameFile159=The Shrine of Mephala by Westly.esp
GameFile160=The Tribe Unmourned (GHD).esp
GameFile161=THTWG - Framerate Version.esp
GameFile162=TLM - NPC Light Sources.esp
GameFile163=Tome of Spellmaking.esp
GameFile164=TravelingMerchants_v2.2.esp
GameFile165=TVwofs.esp
GameFile166=Unboarable Rieklings.esp
GameFile167=Undeads - Skeletons Arise Merged.esp
GameFile168=Unique Ghosts - M - T.esp
GameFile169=UniqueFinery_NoRobe.esp
GameFile170=Vampire Realism II - BM Add-On.esp
GameFile171=Vampire Realism II - TB Add-On.esp
GameFile172=Vampire Realism II.esp
GameFile173=VR_Tweaks.esp
GameFile174=Wanderers of Solstheim.esp
GameFile175=WeatheredSigns.esp
GameFile176=Westly Presents_Half Races of Tamriel_X.esp
GameFile177=Westly Presents_Halflings (nvde).esp
GameFile178=Westly's Master Headpack X.esp
GameFile179=What Thieves Guild.ESP
GameFile180=wl_dockclutter.esp

Just a wild guess based on that list, maybe Less Generic Tribunal has something to do with it?

Oh dear, I hope not, but I'll have to check to find out.
EDIT: Huh, interesting; I unchecked LGNPC Tribunal, went outside, and after killing Gaenor through the console, Azura popped up! It must be being stuck in Combat that's causing this problem.