Breastfeeding isn't just about milk, so that's your first issue assuming you can just replace the source. It's also about comfort and bonding.
The main thing to remember is that it is not your or your wife's job to make him sleep, it's your job to make him feel safe and relaxed enough to be able to fall asleep.
If you're calm and patient, this will pay off. If you're tense, stressed and short tempered he will pick up on those vibes.
Do you manage to put your child down for a nap by yourself? I'd start there, my son was more receptive to that. He spent one day a week at home solo with daddy from the age of 1 when I went back to work, and had no choice but to nap with him.
Dad used to make sure he had a good lunch and drink then cuddle him in the rocking chair for a bit before putting him down either sleepy or asleep. You don't need to rely on milk, and actually at 1 you're supposed to be moving away from bottles and on to cups etc.
Eventually when I stopped breastfeeding (19 months) dad took over bedtime - I would read DS a story then dad would cuddle and put him down. He still wanted mum sometimes, especially if he was poorly or very tired or had a very stimulating day, so it wasn't totally cut and dry.