Are there any famous patriotic/nationalist hymns/verses about Bharatavarsha found in the Vedas, Itihasas or any Hindu scriptures?
Yes, there are several patriotic/nationalist hymns found in Hindu scriptures. For example, here are some Atharva Veda Suktas which are centered towards peace, stability, unity, prosperity of nation and its people. Atharva Veda is full of such patriotic/nationalist hymns. Below are some of the Mantras from Atharva Veda.
Atharva-Veda Kanda 3 - Sukta 8 -राष्ट्र धारण सूक्त.- A charm to secure the submission, love, and fidelity of kinsmen.
This is a patriotic hymn for unification of nation . It's said in the mantra that lets our hearts be one.
अहं गृणभामि मनसा मनांसि मम चित्तमनु चित्तेभिरेत |
मम वशेषु ह्यदयानि व: कृणोमि मम यातमनुवत्र्मान एत || AV 3.8.6||
I with my spirit seize and hold your spirits. Follow with thought and wish my thoughts and wishes. I make your hearts the thralls of my dominion; on me attendant come thy way I guide you.
This mantra is for stability of a Nation.
आ यातु मित्र ऋतुभि: कल्पमान: संवेशयन् पृथिविमुस्त्रियाभि : |
अथास्मभ्यं वरुणो वायुरग्निर्बुहद् राष्ट्रं संवेश्यं दधातु || AV 3.8.1||
Let Mitra come, arranging, with the Seasons, lulling the Earth to rest with gleams of splendour And so let Agni, Varuna, and Vāyu make our dominion tran-quil and exalted.
Here is an English Translation by Ralph T.H. Griffith
Atharva Veda - Kanda 1 - Sukta 29 -Rashta Abhivardhan Sukta - राष्ट्र -अभिवर्धन सूक्त -
This Sukta is for Development of workforce of people for the Prosperity and welfare of Nation.
अभिवर्तेन मणिना येनेन्द्रो अभिवावृधे | तेनास्मान ब्रह्मणस्पतेsभि राष्ट्राय वर्धय || AV 1.29.1 ||
O Brahmanspate ! Develope us for the welfare of the nation through the gem by which Indra flourished.
Atharva Veda - Kanda 6 - Sukta 42 -ParasparaChittaEkikarn Sukta - परस्परचितैकीकरण सूक्त -- Mutual Unity of hearts.
This Sukta is for mutual unity among people of the nation so they can work unitedly as a team by uniting their minds i.e. uniting their minds as one single mind and working towards prosperity welfare and development of nation. The concept integrity among people of a nation is visible in this Sukta which is a good example of patriotic hymns.
सखायाविव सचावहा अव मन्युं तनोमि ते |
अधस्ते अशम्नो मन्युमुपास्यामसि यो गुरु : || AV 6.42.2 ||
By uniting all our minds together , Let us work together with one mind. We discard our anger towards each other ,we crush our anger towards each other with heavy stone.
together let us walk as friends: thy wrathful feeling I remove. Beneath a heavy stone we cast thy wrath away and bury it.
English Translation of the above mantra.
Yes, the Yajurveda contains a famous patriotic hymn in the twenty-second chapter:
आ ब्रह्मन् ब्राह्मणो ब्रह्मवर्चसी जायतामा राष्ट्रे राजन्यः
शूर इषवयोऽअतिव्याधी महारथो जायतां दोग्ध्री धेनुर्वाढानड्वानाशुः सप्तिः
पुरन्धिर्योषा जिष्णू रथेष्ठाः सभेयो युवास्य यजमानस्य वीरो जायतां निकामे निकामे नः
पर्जन्यो वर्षतु फलवत्यो न आषधयः पच्यन्तां योगक्षेमो नः कल्पताम् ॥
English Transliteration: A brahman brAhmaNo brahmavarchasI jAyatAmA rAShTre rAjanyaH
shUra iShavayo&ativyAdhI mahAratho jAyatAM dogdhrI dhenurvADhAnaDvAnAshuH
saptiH purandhiryoShA jiShNU ratheShThAH sabheyo yuvAsya yajamAnasya vIro jAyatAM nikAme nikAme naH
parjanyo varShatu phalavatyo na AShadhayaH pachyantAM yojakShemo naH kalpatAm ||
English Translation: O the Supreme Lord of the world, in this country of ours, let the Brahmins rise as a community of brilliant men of divinity and illustrious scholars of universal knowledge. Let the Kshatriyas rise as a force of heroic warriors of the chariot and the routers of the enemy with unfailing arrows. May the cows be fertile and abundant in milk, the oxen, carriers of heavy load and coursers be swift; the woman, kind and generous; and may the children (people) of the yajamana; the ruler, be brave and fearless, the victorious riders of the chariot, youthful and civil members of the government. May the clouds rain in showers for us according to the need of the season. May the herbs and trees mature and ripen with abundant fruit. And maybe this land be securely blest with a ceaseless cycle of growth and protection of happiness and prosperity. (Yajurveda 22.22)