I do not know the source of the story that was quoted about Nala and Neela in the question. As far as I understood, Valmiki's Srimad Ramayana does not contain this story.
Coming to the question of necessity of praying the Sea God, when Nala was very much available on Sri Rama's side, my answer is follows:
Here we have to understand 2 issues.
1) Construction of a massive bridge for a length of 100 Yojanas, i.e, 100 x 13 kms = 1,300 Kms ( As 1 Yojana = 13 Kms)
2) Permission of Sea God for the passage of Vanaras through sea.
I) Sri Rama started praying the Sea God on the advice of Vibhishana, for safe passage of the army.
उपायैर् अभिगच्चामो यथा नद नदी पतिम् || ६-१९-२९
तराम तरसा सर्वे ससैन्या वरुण आलयम् |
"In what manner shall we along with the army swiftly cross the sea, the lord of streams and rivers, and arrive at that strategy?"
एवम् उक्तस् तु धर्मज्ञः प्रत्युवाच विभीषणः || ६-१९-३०
समुद्रम् राघवो राजा शरणम् गन्तुम् अर्हति |
Hearing their words, the right minded Vibhishana replied as follows: "It is apt if prince Rama seeks ocean as his refuge"
अबद्ध्वा सागरे सेतुम् घोरे अस्मिन् वरुण आलये || ६-१९-३९
लंका न आसादितुम् शक्या स इन्द्रैर् अपि सुर असुरैः |
"Without constructing a bridge across this fiery ocean, which is an abode of Lord Varuna, even the celestials and demons along with Indra cannot reach the city of Lanka."
एवम् उक्तः कुश आस्तीर्णे तीरे नद नदी पतेः |
सम्विवेश तदा रामो वेद्याम् इव हुत अशनः || ६-१९-४२
Having been thus exhorted, Rama then sat on the shore of the ocean covered with blades of Kusha grass, as the God of Fire ascends the altar.
So in order to appease the Sea God, Sri Rama prayed to him.
Then the Sea God, told the way to cross him.
विधास्ये येन गन्तासि विषहिष्ये ह्यहम् तथा |
न ग्राहा विधमिष्यन्ति यावत्सेना तरिष्यति || २-२२-२८
हरीणाम् तरणे राम करिष्यामि यथास्थलम् |
"O, Rama! I shall make it possible to see that you are able to cross over. I will arrange a place for the monkeys to cross me and bear with it. As far as the army crosses me, the crocodiles will not be aggressive to them."
II) Coming to 2nd part of the question, Sri Rama does not know that Nala, the son of Viswakarma, who is on is side, is also great architect as his father is.
It was the Sea God, who told to utilise the services of Nala.
अयम् सौम्य नलो नाम तनुजो विश्व कर्मणः |
पित्रा दत्त वरः श्रीमान् प्रतिमो विश्व कर्मणः || २-२२-४४
"O, excellent man! This one, named Nala, a glorious person, is the son of Vishvakarma; who was given a boon by his father and is equal to Visvakarma."
एष सेतुम् महाउत्साहः करोतु मयि वानरः |
तम् अहम् धारयिष्यामि तथा हि एष यथा पिता || २-२२-४५
"Let this greatly energetic monkey build a bridge across me. I can hold that bridge. He is just the same as his father."