No it won't. Budew starts off at 70 Happiness. It you level it 19 levels while it holds the soothe bell it will have 196 Happiness, assuming it hasn't moved any steps, and hasn't fainted or done anything else to change ts happiness level. Pokemon need at least 220 Happiness to evolve, so it will be close, but not quite there.
Source: I used Psypokes happiness guide, and did the math. You can check my math if you want, but remember the Soothe Bell doubles the happiness values gained.