I understand your confusion! Let's write an example sentence, for illustrative purposes:
Heureusement qu'ils sont là. = (It's a) good thing they're here! How lucky/fortunate that they are (t)here!
First possible confusion: this sentence is not an example of the subjonctif du souhait
You might think of an expression like "Vivement que je sois grand !" that uses the subjunctive. Superficially, "heureusement qu'ils sont là" is similar: both start with an adverb + "que." But the difference is that we could re-write "Que je sois grand vivement (le plus vite possible) !" = "Let me grow up quickly, may I grow up quickly, would that I were all grown up!" It is awkward and unnatural, but in this rewritten form we can see that the sentence is an example of the "subjonctif du souhait," where you use the subjunctive to express a wish or an order (the most famous example being "Qu'ils mangent du gâteau !" = "Let them eat cake!"). Heureusment que... expresses no such order. You could not write "Qu'ils soient là heureusement!"... or if you did, it would mean something entirely different (= "let them be here happily").
Second possible confusion: expressions of personal emotional reaction
It's true that expressions of personal emotional reaction require the subjuntive. Je suis heureuse que tu sois venu. Il est content que tu viennes. And it's true that Heureusement qu'ils sont là expresses an emotional reaction. The problem is that this sentence does not meet the other syntactical requirements for the subjunctive: you must have two clauses with two different subjects, separated by "que": Il est heureux qu'ils soient là = "it is a happy/fortunate/lucky thing that they are here." But when you do not have two clauses with separate subjects, you do not use the subjunctive (unless you are using the subjonctif du souhait to express a desire/order, which as we already established, you are not).
I'm sorry that explanation is a bit confusing. Does it help?
Heureusement is an adverb. There is no reason it should be followed by the subjunctive, unlike the adjective, which may take a subjunctive when used personally or impersonally. Do not be misled by the que, which does not translate into English. It merely links the adverb to the clause. The same sentence could also be written: Heureusement, .... (without que.)
Je suis heureux que tu sois venu.
Il est heureux (impersonal) que tu sois venu.
Heureusement que tu es venu.
Heureusement, tu es venu.