It is good you culture was negative. The ESR or (Erythrocyte Sedimentation Rate) is non specific if it is high:
it can mean autoimmune disease,
arthritis
Inflammation
Tissue death
TB
Since they are probably looking for infections like Tuberculosis (TB). That is why they did the sputum test if your ESR was high perhaps they may ask you to get a Xray to TB skin test (if you haven't been vaccinated with the BCG vaccine). That wouldn't take more than a couple of days. So, that would be the longest hold up I think.
Regards,
Hi,
the ESR is an indicator of inflammatory, the normal rates are around 10-25. I am not sure if it will create a problem with your visa case but they may need more tests. xx