For brushing teeth, I believe they are not supposed to use toothpaste (with fluoride) until older anyway... I still use only a wash cloth as it's also pretty hard to get into my son's mouth to clean his teeth after each meal. I try to play a game first, with 'here comes mr. finger' (with the cloth over it); he laughs all the time at first  and points to his teeth - he knows what's coming - but then fights once I put the finger in his mouth to clean. I try to tell him cleaning top teeth, bottom teeth, back teeth, etc, which helps sometimes. I'm also quite used to his biting my finger while I'm in there, it hurts but tolerable, to get the job done. And yes he'll still fuss and complain, but I just hold him firmly and try to be quick about it. He's fine right after I'm finished, and laughs about it the next time again, when Mr. Finger comes out with a wet cloth.

For nail cutting, distraction is definitely key, so you have to find something extraordinarily fascinating, to keep his focus off the nails. Try TV, if other things don't work? Or get your husband/family to sing and act silly, while you quickly trim the nails, with a baby nail clipper. And sometimes you might need to stop and do it in several tries, if you can't get through all fingers in one go. We find that us singing new songs can really catch his attention for a little while, while trying to get through fussy activities like changing, nail trimming, etc. (Not all the time, but it helps often.)

We brush our 18 months old with a stage 1 tooth brush and toothpaste (fluoride free) when either of us is also brushing our teeth.  When he sees one of us also brushing our teeth then he would open his mouth too.

With nail cutting, I always cut his nails while he's sleeping.  If at night, I would have to use a flashlight.