Because the urinating in broken streams, and for a longer amount of time, is not considered urinating less than usual or not at all. It is considered "pollakiuria" which means urinating in smaller amounts or "dysuria" which means difficult urination.
If he is not urinating at all, or he can not urinate, that would be a medical emergency because that means his kidneys have shut down. That can be a side effect.
But taking longer to urinate and urinating in short streams indicates urinary tract inflammation rather than kidney disease. When the kidneys are not working, they stop making any urine. So he would not try to urinate because there would be no urine to come out.
This is from the Plumbs veterinary drug handbook which is what we always use.
Adverse effects with the cephalosporins are usually not serious and have a relatively low frequency of occurrence.
In addition to the adverse effects listed below, cephalexin has reportedly caused salivation, tachypnea and excitability in dogs, and emesis and fever in cats. Nephrotoxicity occurs rarely during therapy with cephalexin, but patients with renal dysfunction, receiving other nephrotoxic drugs or that are geriatric may be more susceptible. Interstitial nephritis, a hypersensitivity reaction, has been reported with many of the cephalosporins including cephalexin. The incidence of these effects is not known.
Hypersensitivity reactions unrelated to dose can occur with these agents and can manifest as rashes, fever, eosinophilia, lymphadenopathy, or full-blown anaphylaxis. The use of cephalosporins in patients documented to be hypersensitive to penicillin-class antibiotics is controversial. In humans, it is estimated 1-15% of patients hypersensitive to penicillins will also be hypersensitive to cephalosporins. The incidence of cross-reactivity in veterinary patients is unknown.
When given orally, cephalosporins may cause GI effects (anorexia, vomiting, diarrhea). Administering the drug with a small meal may help alleviate these effects. Because the cephalosporins may also alter gut flora, antibiotic-associated diarrhea or proliferation of resistant bacteria in the colon can occur.
Rarely, cephalexin has been implicated in causing toxic epidermal necrolysis in cats.
While cephalosporins (particularly cephalothin) have the potential for causing nephrotoxicity at clinically used doses in patients with normal renal function, risks for the occurrence of this adverse effect appear minimal.
High doses or very prolonged use has been associated with neurotoxicity, neutropenia, agranulocytosis, thrombocytopenia, hepatitis, positive Coomb's test, interstitial nephritis, and tubular necrosis. Except for tubular necrosis and neurotoxicity, these effects have an immunologic component
So I think you are seeing urinary tract signs rather than signs of the kidneys failing. Nephrotoxic means kidneys. Other signs of kidney disease are vomiting and lack of appetite.