Gothic was the first language that J.R.R. Tolkien studied for his own pleasure, and it may have given an impulsion in the first development of Qenya. Tolkien even attempted to reconstruct some parts of the language and such elements survived in Taliska, the language he created for the Edain of the First Age. He also composed a full poem, Bagme Bloma "The Flower of the Trees" in Gothic.
Gothic names in Tolkien's legendarium
Since in The Lord of the Rings Old English represents Rohirric, Tolkien used Gothic to represent the earlier language of the Northmen of Rhovanion in accordance to the similarity between these languages.
However, in reality Old English was not directly related to Gothic as the way Rohirric was to Northern; therefore Tolkien's simulation does not reflect exactly the relationship between these languages but rather their similarity.