
According to what I have seen in the literature so far, the standard procedure consists of two main steps:
Prove the existence of a universal $ C^{*} $algebra $ A_{\theta} $ generated by two unitaries $ u $ and $ v $ that satisfy
$$
u v = e^{2 \pi i \theta} v u.
$$
Note: We are assuming that $ \theta $ is irrational.
Prove that $ A_{\theta} $ is simple, and conclude that the concrete realization given above is indeed universal.
To accomplish Step 1, there are several methods. I understand that you have read Davidson’s book, so let me describe an approach different from his that is more algebraic in nature.
Let $ \mathcal{P} $ denote the free associative unital $ \mathbb{C} $algebra in four indeterminates, $ u $, $ u^{*} $, $ v $ and $ v^{*} $, where the identity element of $ \mathcal{P} $ is denoted by $ \mathbf{1} $.
Let $ \mathcal{I}_{\theta} $ denote the (not a priori proper) twosided ideal
$$
\left\langle
u u^{*}  \mathbf{1},
u^{*} u  \mathbf{1},
v v^{*}  \mathbf{1},
v^{*} v  \mathbf{1},
u v  e^{2 \pi i \theta} v u
\right\rangle.
$$
Form the quotient $ \mathbb{C} $algebra $ \mathcal{A}_{\theta} \stackrel{\text{df}}{=} \mathcal{P} / \mathcal{I}_{\theta} $.
Let $ \dot{\mathbf{1}} $, $ \dot{u} $, $ \dot{u}^{*} $, $ \dot{v} $ and $ \dot{v}^{*} $ denote the images of $ \mathbf{1} $, $ u $, $ u^{*} $, $ v $ and $ v^{*} $ in $ \mathcal{A}_{\theta} $ respectively.
Then the following monomials are distinct in $ \mathcal{A}_{\theta} $ and define a Hamel basis for it:
$$
\dot{\mathbf{1}}, \quad
\dot{u}^{m} \dot{v}^{n}, \quad
\dot{u}^{m} (\dot{v}^{*})^{n}, \quad
(\dot{u}^{*})^{m} \dot{v}^{n}, \quad
(\dot{u}^{*})^{m} (\dot{v}^{*})^{n}; \qquad
(m,n) \in \mathbb{N}_{0}^{2} \setminus \{ (0,0) \}.
$$
As such, $ \mathcal{I}_{\theta} $ is a proper ideal of $ \mathcal{P} $, and so $ \mathcal{A}_{\theta} $ is a unital $ \mathbb{C} $algebra.
Comment: WeggeOlsen claims that Point (5) is obvious, but I beg to differ as its validity depends on a nontrivial algebraic result called the Diamond Lemma for Ring Theory. Davidson appears to avoid all forms of algebraic machinery by resorting to the GNS Construction. However, as a staunch believer in the Principle of Conservation of Difficulty, I think that Davidson is simply transferring all technical issues from the Diamond Lemma to the GNS Construction, which, as most operator algebraists would agree, is a highly nontrivial result in the representation theory of $ C^{*} $algebras.
Define a $ C^{*} $representation of $ \mathcal{A}_{\theta} $ to be a triple $ (A,s,t) $, where:
 $ A $ is a unital $ C^{*} $algebra.
 $ s $ and $ t $ are unitary elements of $ A $ satisfying $ s t = e^{2 \pi i \theta} t s $.
Given a $ C^{*} $representation $ (A,s,t) $ of $ \mathcal{A}_{\theta} $, there exists a unique unital $ \mathbb{C} $algebra homomorphism from $ \mathcal{P} $ to $ A $ defined by
$$
u \mapsto s, \quad
u^{*} \mapsto s^{*}, \quad
v \mapsto t \quad \text{and} \quad
v^{*} \mapsto t^{*}.
$$
Then as the homomorphism kills $ \mathcal{I}_{\theta} $, we obtain a unital $ \mathbb{C} $algebra homomorphism $ \pi_{A,s,t}: \mathcal{A}_{\theta} \to A $, once again unique, that satisfies:
 $ {\pi_{A,s,t}}(\dot{u}) = s $ and $ {\pi_{A,s,t}}(\dot{u}^{*}) = s^{*} $.
 $ {\pi_{A,s,t}}(\dot{v}) = t $ and $ {\pi_{A,s,t}}(\dot{v}^{*}) = t^{*} $.
 $ {\pi_{A,s,t}}(\dot{u}) ~ {\pi_{A,s,t}}(\dot{v}) = e^{2 \pi i \theta} ~ {\pi_{A,s,t}}(\dot{v}) ~ {\pi_{A,s,t}}(\dot{u}) $.
Question: Do $ C^{*} $representations of $ \mathcal{A}_{\theta} $ exist?
Answer: Yes! The concrete realization $ (\mathscr{B}({L^{2}}(\mathbb{T})),U,V) $ described by the OP is one. Amusingly, the very existence of this concrete realization shows that $ \mathcal{I}_{\theta} $ is a proper twosided ideal of $ \mathcal{P} $, but I think we can safely say that this is not a demonstration of the fact from first principles.
Define a mapping $ \ \cdot \_{0}: \mathcal{A}_{\theta} \to [0,\infty] $ by
$$
\ a \_{0} \stackrel{\text{df}}{=}
\sup
(\{
\ {\pi_{A,s,t}}(a) \_{A} \in \mathbb{R}_{\geq 0} \mid
\text{$ (A,s,t) $ is a $ C^{*} $representation of $ \mathcal{A}_{\theta} $}
\})
$$
for each $ a \in \mathcal{A}_{\theta} $.
Proof sketch that $ \ \cdot \_{0} $ is a $ \mathbb{C} $algebra seminorm:
 Thanks to the existence of a $ C^{*} $representation of $ \mathcal{A}_{\theta} $, we have
$$
\left\ \dot{\mathbf{1}} \right\_{0}
= \ \dot{u} \_{0}
= \ \dot{u}^{*} \_{0}
= \ \dot{v} \_{0}
= \ \dot{v}^{*} \_{0}
= 1.
$$
 As $ \dot{\mathbf{1}} $, $ \dot{u} $, $ \dot{u}^{*} $, $ \dot{v} $ and $ \dot{v}^{*} $ generate $ \mathcal{A}_{\theta} $, it follows that $ \ a \_{0} < \infty $ for each $ a \in \mathcal{A}_{\theta} $.
 Knowing now that $ \ \cdot \_{0}: \mathcal{A}_{\theta} \to [0,\infty) $, it is easily shown to satisfy the axioms of a $ \mathbb{C} $algebra seminorm.
Notice that $ \ \cdot \_{0} $ also satisfies the $ C^{*} $identity.
Let $ \mathcal{N} \stackrel{\text{df}}{=} \{ a \in \mathcal{A}_{\theta} \mid \ a \_{0} = 0 \} $. Then $ \mathcal{N} $ is a $ \mathbb{C} $subalgebra of $ \mathcal{A}_{\theta} $.
Form the quotient $ \mathbb{C} $algebra $ \mathcal{A}_{\theta} / \mathcal{N} $ to get a pre$ C^{*} $algebra, denoting the quotient norm by $ \ \cdot \ $.
Complete $ \mathcal{A}_{\theta} / \mathcal{N} $ with respect to $ \ \cdot \ $ to obtain the irrational rotation $ C^{*} $algebra $ A_{\theta} $.
Clearly, $ A_{\theta} $ is unital and is generated by the pair $ ([\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}) $ of unitary elements. Furthermore,
$$
[\dot{u}]_{\mathcal{N}} [\dot{v}]_{\mathcal{N}}
= e^{2 \pi i \theta} [\dot{v}]_{\mathcal{N}} [\dot{u}]_{\mathcal{N}}.
$$
This completes the construction.
Claim: $ (A_{\theta},[\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}) $ is a universal $ C^{*} $representation of $ \mathcal{A}_{\theta} $.
Proof of Claim
Let $ (A,s,t) $ be a $ C^{*} $representation of $ \mathcal{A}_{\theta} $. Then by the definition of $ \ \cdot \_{0} $,
$$
\forall a \in \mathcal{A}_{\theta}: \quad
\ {\pi_{A,s,t}}(a) \_{A} \leq \ a \_{0} = \ [a]_{\mathcal{N}} \.
$$
We thus have a unique unital $ * $homomorphism $ \pi_{A,s,t}^{\mathcal{N}}: \mathcal{A}_{\theta} / \mathcal{N} \to A $ satisfying
$$
{\pi_{A,s,t}^{\mathcal{N}}}([\dot{u}]_{\mathcal{N}}) = s \quad \text{and} \quad
{\pi_{A,s,t}^{\mathcal{N}}}([\dot{v}]_{\mathcal{N}}) = t,
$$
and we can extend this, using continuity, to a unique unital $ * $homomorphism $ \Pi_{A,s,t}: A_{\theta} \to A $. In other words, $ \Pi_{A,s,t} $ is the only unital $ * $homomorphism from $ A_{\theta} $ to $ A $ that maps $ [\dot{u}]_{\mathcal{N}} $ to $ s $ and $ [\dot{v}]_{\mathcal{N}} $ to $ t $.
Now, suppose that there is another unital $ C^{*} $algebra $ B $ generated by two unitaries $ u' $ and $ v' $ satisfying
$$
u' v' = e^{2 \pi i \theta} v' u'
$$
such that for any $ C^{*} $representation $ (A,s,t) $, there exists a unique $ * $homomorphism $ \Phi_{A,s,t}: B \to A $ that maps $ u' $ to $ s $ and $ v' $ to $ t $.
The following statements are then true:
 $ \Pi_{B,u',v'}: A_{\theta} \to B $ is the unique $ * $homomorphism that maps $ [\dot{u}]_{\mathcal{N}} $ to $ u' $ and $ [\dot{v}]_{\mathcal{N}} $ to $ v' $.
 $ \Phi_{A_{\theta},[\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}}: B \to A_{\theta} $ is the unique $ * $homomorphism that maps $ u' $ to $ [\dot{u}]_{\mathcal{N}} $ and $ v' $ to $ [\dot{v}]_{\mathcal{N}} $.
 $ \Phi_{A_{\theta},[\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}} \circ \Pi_{B,u',v'}: A_{\theta} \to A_{\theta} $ equals $ \text{id}_{A_{\theta}} $ on a dense subset of $ A_{\theta} $.
 $ \Pi_{B,u',v'} \circ \Phi_{A_{\theta},[\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}}: B \to B $ equals $ \text{id}_{B} $ on a dense subset of $ B $.
Therefore, $ \Phi_{A_{\theta},[\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}} \circ \Pi_{B,u',v'} = \text{id}_{A_{\theta}} $ and $ \Pi_{B,u',v'} \circ \Phi_{A_{\theta},[\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}} = \text{id}_{B} $, and so $ A_{\theta} $ is $ * $isomorphic to $ B $ via the unique $ * $isomorphism that sends $ [\dot{u}]_{\mathcal{N}} $ to $ u' $ and $ [\dot{v}]_{\mathcal{N}} $ to $ v' $.
This concludes the proof that $ (A_{\theta},[\dot{u}]_{\mathcal{N}},[\dot{v}]_{\mathcal{N}}) $ is indeed a universal $ C^{*} $representation of $ \mathcal{A}_{\theta} $. $ \quad \blacksquare $
The proof that $ \Pi_{\mathscr{B}({L^{2}}(\mathbb{T})),U,V}: A_{\theta} \to {C^{*}}(U,V) $ is a $ * $isomorphism follows from the nontrivial fact that $ A_{\theta} $ is a simple $ C^{*} $algebra (i.e., it has no nontrivial closed proper twosided ideals). The main idea behind the proof is to use the socalled trace function on $ A_{\theta} $.
This trace function does wonders for us. Firstly, it shows that $ A_{\theta} $ contains a nontrivial projection element. Secondly, it shows that $ \mathcal{A}_{\theta} $ is faithfully represented as a $ \mathbb{C} $algebra in $ A_{\theta} $, i.e., $ \mathcal{N} = \{ 0_{\mathcal{A}_{\theta}} \} $. Observe that in defining the $ C^{*} $seminorm $ \ \cdot \_{0} $, there was no guarantee that each nonzero element of $ \mathcal{A}_{\theta} $ would not be sent by $ \ \cdot \_{0} $ to $ 0 $. Playing around with the trace function shows that this is indeed the case.
