Both items, explained

Let's start by looking what both items give. We picked up a Philosopher's Stone early game, which gives us some neat early game gold and HP/5 and MP/5.

Philosopher's StonePhilosopher's Stone 7 Health Regeneration, 9 Mana Regeneration. UNIQUE Passive: Transmute: Gain +5 additional gold every 10 seconds. Costs: 700 gold, 520 gold upgrade from Faerie Charm.
Now, we've got these two upgrades:

Eleisa's Miracle 10 Health Regeneration, 15 Mana Regeneration. UNIQUE Passive: Aid: Your Summoner Heal, Revive, and Clairvoyance cooldowns are reduced by 20%. Eleisa's Blessing: If you gain 3 levels with this item, you gain all the effects of this item permanently and this item is consumed. Costs: 1100 gold, 400 gold upgrade from Philosopher's Stone.
Shurelya's Reverie 250 Health, 10 Health Regeneration, 10 Mana Regeneration. UNIQUE Passive: +10% Cooldown Reduction. UNIQUE Active: Nearby allies gain 40% Movement Speed for 3 seconds (600 range). 60 second cooldown. Costs: 2100 gold, 1400 gold upgrade from Philosopher's Stone.
Now, since Eleisa's Miracle is being consumed after three levels, we know that you will have a steady +10 HP/5 and +15 MP/5. The UNIQUE Passive: Transmute is only useful when you are using these spells ( Heal, Revive and Clairvoyance). Since I normally don't like to run Heal (because you have a good heal of yourself, see the "Spells" section about this), I don't benefit from this passive. If you do use Heal instead of Exhaust, this is of course a brilliant passive.

Reasoning

When you choose not to pick up Eleisa's Miracle but take Shurelya's Battlesong instead, your added stats are +250 HP, +10 HP/5, +10 MP/5 and +10% CDR, instead of +10 HP/5 and +15 MP/5. So we trade +5 MP/5 for 250 HP and +10% CDR. We also have a good active for quick retreats and quick engages.

The problem is we have to drop an item out of our build. While Eleisa's Miracle is being consumed after leveling up three times, Shurelya's Battlesong will keep occupying the slot, thus losing one of the following list of stats:

+300 HP, Wards +45 AP, +4 GP/10, AURA: +6 MP/5, Slowing active +40 MR, +300 MP, +9 MP/5, MP/5 increase based on missing MP, Built-in Cleanse for allies +20 Armor, +400 HP, +30 MR, AURA: +10 Armor, AURA: +30 MR, AURA: +10 HP/5
Since you'll have to ward as a support, dropping Ruby Sightstone is not an option. Period.
When you drop Shard of True Ice, you'll lose GP/10. You'll be benifiting from the GP/10 from Philosopher's Stone longer though.. Let's say you keep Philosopher's Stone for 10 more minutes. That's 10 (minutes) X 6 (60 seconds per minute, you'll get gold per 10 seconds) X 1 (extra gold per 10 seconds) = 60 gold. For me, that doesn't cut losing 45 AP (which is extra heal, for instance), a 6 MP/5 aura (which makes you lose 11 MP/5 already, 6 from this item and 5 from Eleisa's Miracle) and a good slowing active in favor of a movement speed active. You do get +10% CDR and +250 HP. But wait... there's more!

Let's look at that CDR. The cooldown on Ebb and Flow is 9 seconds when maxed. It will heal you for 230 HP + 50% of ability power. Let's take a look what happens when we spam this skill for 3 minutes. When you've got no extra CDR or AP, you can cast this skill exactly 20 times. I'll heal my ally for 230 HP each time (I only count first hit on the ally, bear with me please) which will total for 4600 HP healed. Let's say we go with +10% CDR, and look at spamming the skill for 3 minutes. You'll be using this skill for 22 times (rounded down) and will do 230 HP recovery each time... You'll heal for 5060 HP total. Now, without the CDR but with 45 AP, I can cast Ebb and Flow 20 times in 3 minutes, healing for 230 + 22.5 (50% of 45 AP) a pop, which totals 5050 HP. 10 HP less healing, while losing more MP/5, damage, gold in favor of +250 HP and a slowing active... Not an option for me.
Okay, let's say we drop Mikael's Blessing. Now, we know this build has been made for spam. You want to use Nami's Ebb and Flow and Tidecaller's Blessing as much as possible, either for healing, damage, slowing or movement speed buff from your passive Surging Tides. These things cost mana, and MP/5 is really crucial on Nami

When we drop Mikael's Blessing, we not only lose +5 MP/5 from Eleisa's Miracle, but also the +9 MP/5 from this item. Furthermore, we lose the UNIQUE Passive: Mana Font from this item, which increases MP/5 by 1% per 1% of missing MP. You basically can't use your abilities all the time, and have to play mana-conserving, which will eventually put you behind and you'll be significantly less useful to the team.

Furthermore: we lose a brilliant active. The active on Mikael's Blessing removes all CC on an ally and heals them for 150 HP + 15% of their missing health. Even though there is a 3 minute cooldown on this active (which is lowered by masteries, but still, it's long), this is such a life-saving active in case a teamfight goes wrong. When you're AD carry is being put under a lot of CC and is focused down in a big teamfight, you'll want to use this. It will change the complete outcome of that teamfight. Very valuable item, not an option to drop in favor of Shurelya's Battlesong.
Final option: Runic Bulwark. Except the difference of -5 MP/5 when choosing for Shurelya's Battlesong over Eleisa's Miracle, you'll also lose 150 HP ( Shurelya's Battlesong gives you +250 HP where this item would give you +400 HP). We will get +10% CDR, which combined with the 45 AP from Shard of True Ice would let me heal for 5555 HP (505 HP more thanks to +10% CDR, see calculations a little above).

This item does more, though. It'll give you +20 Armor and +30 MR, which makes you survive longer. But more importantly, your nearby allies will lose +10 Armor, +30 MR and +10 HP/5 from the aura of Runic Bulwark. You'll be healing them for 505 HP more over 3 minutes, which is a rough 8.4 HP/5 (rounded down). Though the movement speed buff is nice, personally I wouldn't want to lose the defensive stats Runic Bulwark gives me. So no, no deal.

Then there are the situational picks... For the AP reasons already declared above, Banned of Command and Twin Shadows are no good options either. Ironically, Twin Shadows actually gives yourself (not the team, only you) a speed buff and actually can slow enemy champions, partially achieving the same result as with the active on Shurelya's Battlesong.

With dropping Zeke's Herald you'll lose more then you get in return: -5 MP/5 (from Eleisa's Miracle), -5% CDR, you won't get health extra (you get +250 HP from Shurelya's Battlesong, but you wont get the +250 HP from Zeke's Herald, which effectively renders this +0 HP), and you and nearby allies loes +20 AD and +10% Lifesteal from your aura, in favor of a movement speed buff active... You don't have to be a rocket scientist to see that is a bad trade. No deal.

Which leaves Locket of the Iron Solari. No extra CDR, -35 Armor, only -175 HP, -5% MP/5 (from Eleisa's Miracle) and a movement speed active in favor of a 230HP (when level 18) shield on all nearby allies for 5 seconds. Again, not a good trade.

So, when do we want to pick Shurelya's Battlesong?

In the rare occasion you absolutely 100% need a quick in or a quick out. Let's say you win a teamfight, and you push mid lane like there is no tomorrow. The enemy team respawns while your team is still hacking and slashing on the inhibitor and since you know the enemy team has such a bunch of CC covering your entire (or great part) of the team, you know you wont make it out alive (and thus, lose the game) unless you run like a mad man! When you know that's going to happen: buy Shurelya's Battlesong. Use the active wisely, and it will pay off the bad trades you'll have to make which I outlined above. Otherwise: forget it. It's not worth the item slot.