I was in a similar situation like one year ago. And I had to learn NMA models and BUGS at the same time. However, I think better way is starting with pairwise meta-analysis (conventional meta-analysis) with WinBUGS, and then going to NMA models which are more complicated. Actually a pairwise meta-analysis is just a network meta-analysis with only two treatments. The BUGS tutorials has one example with a pairwise meta-analysis which you can read. Or you can read any other WinBUGS tutorial and apply to the example I just mentioned. And it may take some time, because you need to have some knowledge about Bayesian inference (prior and posterior distributions etc.), Markov Chain Monte Carlo (checking convergence diagnostics etc.)

Once you learned how WinBUGS work, then usually the only thing you need is the right BUGS code. NICE reports are good resources for NMA models using WinBUGS. More specifically, after you learned how to fit pairwise meta-analysis using WinBUGS, you can read this technical report. In this report, the BUGS codes for different type of NMA models are shown and explained in detailed.