With DS1 I was much the same as you, hitched a ride in a wheel chair with a blanket covering my bare bottom LOL. That's the only reason, I was a mess all over & only had my top to put back on. But soon after I was in my room & had a shower I walked DS down to the main area to bath him at just a few hours old.
With DS2 I had him in about 20minutes, showered up in the ward then held onto his cot (for balance) and pushed him back to our room....was a tad wobblely due to the shock of it all happening so very quickly.