Which two poems best exemplify the characteristics of British Romanticism and Romantic poetry?
This list can be quite exhausting. There were so many poems that were produced during the Romantic time period that were so highly conscious of following the forms and ideals of Romanticism that it could be quite open in selection of poems. For example, if I were to select a poem that emphasized the power of the natural setting, Wordsworth's "I Wandered Lonely as a Cloud" or Shelley's "Ode to the West Wind" could be two of literally hundreds that could be chosen. At the same time, if I wanted to select a poem that emphasized the individual experience and mode of subjectivity as being essential to consciousness, I could select Wordsworth's "The Solitary Reaper" or Byron's "She Walks in Beauty." In the end, identifying the specific characteristics that are being analyzed can help out greatly in selecting matching poems.
There really aren't just two poems that exemplify Romanticism in Britain. There are many. I can give you two that do it very well, though.
Wordsworth's "Tintern Abbey" demonstrates the romantic view of nature, its possible effects on human imagination and character, and Wordsworth's writing process. He demonstrates how he experiences nature, recollects it, then writes about it.
Shelley's "Ode to the West Wind" also reveals the romantic idea of nature working in human imagination--Shelley desires that the wind help him to do for poetry in the world what the wind does to the landscape.
Again, Romanticism is multi-faceted and is represented by a multitude of poems. These are just two that demonstrate romantic characteristics.