I have had time now to level my Paladin team and now have had some experience with healing with the Priest and healing with the Holy Paladin.

A few details about the team: the priest and blood DK's are ilvl 880 (give or take 1-2 points). They each have 2-3 Legendaries (they all have a crafted plus 1-2 others). The Holy Paladin is ilvl 865 with the crafted Legendary belt.

Here are my thoughts, along with the obligatory "YMMV":

The Holy Priest is a very strong group healer. So many good options for AoE healing, along with fairly strong single target healing options. A lot of the healing spells are instant-cast so movement is less of an issue. I personally do not care for the artifact talent/spell, but that is probably just me. However, the Hpriest is just so squishy. Even on Mythic+1, the boss abilities can sometimes one-shot the Hpriest. I am not familiar enough with some of the fights to be able to prepare the priest for incoming damage and even when I know the damage is incoming, the Hpriest just does not seem to have a lot of mitigation spells and gets fried.

The Discipline Priest is fun to play. There is much more to do with respect to healing and damage dealing. The DPS of the Dpriest translates to self or group healing, depending on who has the Atonements. Group healing is good, although not as good as the Holy Priest, and movement fights are rough as a lot of the healing spells and damage dealing spells are 1-2 seconds casts. Power Word: Shield is a useful spell for incoming damage (and also applies Atonement) which can improve survivability. However, the Priest in Discipline form is still very squishy and will still get one-shot.

The Holy Paladin has been a completely different experience in many ways. Single target and group healing are both very good. There is a good mixture of instant-cast and short cast time healing spells. The Hpaladin also has a number of useful "oh shit" spells (LoH, Divine Shield) to react to sudden damage. The Hpaladin is also less squishy. However, since a lot of the damage abilities and spells are melee range only, DPS suffers due to DPS requiring more button mashing and slows my reaction to boss abilities.

Honestly, I think that all of these options would potentially be viable if I get more experience under by belt. However, I think that the Holy Paladin is the best option for my playstyle followed by the Discipline Priest. I am still going to give the Dpriest a shot, but for the time being I am going to get the Paladin team to the same point with respect to class hall and Suramar quests (and Azshara, as that was the last zone in which I leveled the paladin team) and focus on the Holy Paladin as healer for the DK's.