Urate oxidase (UOx) surrounded by synthetic macromolecules, such as polyethyleneimine (PEI), poly(allylamine hydrochloride) (PAH), and poly(sodium 4-styrenesulfonate) (PSS) is a convenient model of redox-active biomacromolecules in a crowded environment and could display high enzymatic activity towards uric acid, an important marker of COVID-19 patients. In this work, the carbon fiber electrode was modified with Prussian blue (PB) redox mediator, UOx layer, and a layer-by-layer assembled polyelectrolyte film, which forms a complex coacervate consisting of a weakly charged polyelectrolyte (PEI or PAH) and a highly charged one (PSS). The film deposition process was controlled by cyclic voltammetry and scanning electron microscopy coupled with energy-dispersive X-ray analysis (at the stage of PB deposition) and through quartz crystal microbalance technique (at latter stages) revealed uniform distribution of the polyelectrolyte layers. Variation of the polyelectrolyte film composition derived the following statements. (1) There is a linear correlation between electrochemical signal and concentration of uric acid in the range of 10−4–10−6 M. (2) An increase in the number of polyelectrolyte layers provides more reproducible values for uric acid concentration in real urine samples of SARS-CoV-2 patients measured by electrochemical enzyme assay, which are comparable to those of spectrophotometric assay. (3) The PAH/UOx/PSS/(PAH/PSS)2-coated carbon fiber electrode displays the highest sensitivity towards uric acid. (4) There is a high enzyme activity of UOx immobilized into the hydrogel nanolayer (values of the Michaelis–Menten constant are up to 2 μM) and, consequently, high affinity to uric acid.