Alyansı görünce gelin rahatladı; nikâh birazdan başlayacakmış.

Because -ı here is the accusative case ending.

• alyans = wedding ring / wedding band
• alyansı = the wedding ring as a specific direct object

So Alyansı görünce means when she saw the wedding ring.
Without the accusative, alyans görünce would sound less natural here and would not clearly mean a specific ring.

In this sentence, görünce comes from görmek = to see.

It is built like this:

• gör- = see
• -ünce = a converb suffix meaning when / once / upon

So:

• gör + ünce = görünce = when (someone) sees / saw

This is a very common learner question because görünce looks like it might come from görünmek = to appear / be seen, but here it does not. Context shows it means when she saw.

This suffix usually means when, once, after, or upon doing something.

In this sentence:

• Alyansı görünce gelin rahatladı
  = When the bride saw the ring, she relaxed

It is mainly a time relationship: one thing happens, then the main action happens.

Depending on context, it can also feel a bit causal in English:

• Seeing the ring made her feel relieved

But grammatically, the Turkish form is basically a when/once structure.

Because in Turkish, a clause with -ince often has an understood subject, and that subject is usually the same as the subject of the main clause unless something else is made clear.

Here the main clause is:

• gelin rahatladı = the bride relaxed

So the natural reading is:

• When the bride saw the ring, the bride relaxed

Turkish does not need to repeat the subject if it is clear from context.

Rahatladı is the simple past of rahatlamak.

• rahatlamak = to relax, to feel relieved, to become at ease
• rahatladı = relaxed / felt relieved / became relieved

In this sentence, felt relieved is probably the best natural English sense, because it suggests she had been worried and then stopped worrying.

Because Turkish has no articles like a/an/the.

So:

• gelin can mean a bride or the bride
• nikâh can mean a wedding ceremony or the wedding ceremony

Context tells you which one is meant.
In this sentence, both are clearly definite in English:

• the bride
• the wedding ceremony

Not exactly.

• nikâh = the marriage ceremony itself, especially the formal act of getting married
• düğün = the broader wedding celebration, including the party, guests, music, etc.

So nikâh birazdan başlayacakmış means:

• the marriage ceremony was apparently going to start soon

not necessarily the whole wedding celebration.

Birazdan means in a little while, shortly, or soon.

It refers to the near future.
So:

• Nikâh birazdan başlayacakmış
  = The ceremony was apparently going to start soon

It is very close in meaning to biraz sonra, and in many situations they are interchangeable.

Because -mış adds an extra meaning beyond simple future.

Breakdown:

• başla- = start
• -yacak = future
• -mış = reported / inferred / apparently

So başlayacakmış can mean things like:

• apparently it will start soon
• they said it would start soon
• it turned out it was going to start soon

The speaker is not presenting the information as a plain direct statement.
There is some sense of hearsay, discovery, or indirect knowledge.

Either can be right, depending on the wider context.

Grammatically, başlayacakmış is still a future form, but the future is relative to some reference point.

So:

• if the whole story is being told from a past point of view, English often says
  was going to start soon or was about to start
• if the speaker is talking more directly about the future, English might say
  will apparently start soon

In many real translations, English does not always show the -mış nuance very explicitly, even though Turkish does.

Because the subject is nikâh, which is third person singular, and in Turkish third person singular often has no extra personal ending.

So:

• ben başlayacağım = I will start
• sen başlayacaksın = you will start
• nikâh başlayacak = the ceremony will start

The same idea applies in başlayacakmış.
The subject nikâh already tells you who or what is starting.

Yes, that would also be natural.

Turkish word order is flexible, although the verb usually comes at the end. Different orders can shift focus slightly.

• Alyansı görünce gelin rahatladı puts a little more attention on the ring
• Gelin alyansı görünce rahatladı starts with the bride, which may feel more neutral to some learners

Both are grammatical. The sentence you were given is completely natural.