You gain Ability Score Improvements (ASIs) as a multiclass character at the following levels: 4, 8, 12, 16, 19, rather than when you reach a particular level in a class.
If you have class levels in two classes which gain the Extra Attack feature, you gain the Extra Attack feature at level 5 (or the highest level between the two). This allows the Warlock to gain the Thirsting Blade invocation sooner.
The Fighter and Rogue have additional bonus ASIs. When you obtain a level in one of these two classes that provides an extra ASI, you gain it then. You do not gain these simply for having a level in one of these two classes. You must obtain the level of these extra bonus ASIs.
An Eldritch Knight 14/Arcane Trickster 6 has 20 levels as a 1/3 per level spells per day multiclass split. The rogue levels count for 2 levels of spellcasting, and the EK counts for 4 2/3 per day. In this scenario, the character has 6 levels of spellcasting. With this proposed variant, you would take all the classes with the same fraction together first (1:1 for Bard/Cleric/Druid/Sorcerer/Wizard, 1:2 for Paladin/Ranger, 1:3 for Eldritch Knight/Arcane Trickster) and calculate the total. This would call the character to be recognized as having 20 levels / 6, which is still 6 2/3, but the 2/3 is almost ready for an upgrade and could receive one from Paladin/Ranger.
The purpose in this is trying to not cause a player to lost gained levels in their stat calcuations. An EK 5/Ranger 3 for instance would have 1-2/3 spellcasting from EK and 1-1/2 from Ranger. Normally, the character would be considered to have two levels of casting (1-2/3 rounded down = 1, 1-1/2 rounded down = 1; 1 + 1 = 2). In this variant, the character would now have 3 (1-2/3 + 1-1/2 = 3-1/6 casting levels).