Charlotte Christodoulou from Walsall, West Midlands, was discovered in the in the grounds of Harlescott Junior school on 4 January.
Her grandfather Pete said the "fun-loving" 23-year-old was a "precious part" of the family.
West Mercia Police said they were no longer treating her death as murder.
An initial post-mortem examination proved inconclusive and further tests are being carried out.
Three people arrested on suspicion of obstructing a coroner have been released on police bail until 6 March.
The family of a woman found dead at a school in Shrewsbury have paid tribute to "a wonderful mother, daughter, sister, niece and granddaughter".