groupby and find the first non-zero value

You can use GroupBy.transform with a custom function to get the index of the first non-zero value with idxmax (that return the first True value here):

df['desired_first_nonzero'] = (df.groupby('group')['value']
                                 .transform(lambda s: s[s.ne(0).idxmax()])
                              )

alternatively, using an intermediate Series:

s = df.set_index('group')['value']
df['desired_first_nonzero'] = df['group'].map(s[s.ne(0)].groupby(level=0).first())

output:

   group     date  value  desired_first_nonzero
0      1  jan2019      0                      2
1      1  jan2019      2                      2
2      1  feb2019      3                      2
3      1  mar2019      4                      2
4      1  mar2019      5                      2
5      2  feb2019      0                      4
6      2  feb2019      0                      4
7      2  mar2019      0                      4
8      2  mar2019      4                      4
9      2  apr2019      5                      4

This should do the job:

# the given example
d = {'group': [1, 1, 1, 1, 1, 2, 2, 2, 2, 2], 'value': [0, 2, 3, 4, 5, 0, 0, 0, 4, 5]}
df = pd.DataFrame(data=d)

first_non_zero = pd.DataFrame(df[df['value'] != 0].groupby('group').head(1))
print(first_non_zero)

Output:

    group   value
1   1   2
8   2   4

Then you can distributed as needed for each group row.