As I wrote in my comment above, the OP is about two different classes of groups: 2-generated universal countable groups (these contain all countable groups and are not finitely presented) and universal finitely presented group (these contain all recursively presented groups and are finitely presented). The question about minimal number of relators makes sense for the second class only. The only known lower bound is 1 (1-related groups have solvable word problem and are not universal). Every universal finitely presented group has unsolvable word problem. I do not think there is an example of a universal group with few relators. The state of the art of few-relator groups with unsolvable word problem is in Collins paper of 1986: https://projecteuclid.org/download/pdf_1/euclid.ijm/1256044631. The smallest known presentation has 12 relators.
Panov, D. , Petrunin, A.
The telescopic construction: a microsurvey. Proceedings of the Gökova Geometry-Topology Conference 2013, 110–119, Gökova Geometry/Topology Conference (GGT), Gökova, 2014.
for a different notion of universality of f.p. groups.