William is probably the one you are looking for.
In 1861 Elizabeth Maddrell 37 b Lonan, is living at 12 Queen Street Douglas with children Emma 16, Joseph 10, William 8, Louis A 3 and George 8 months, all born in Douglas as well as her parents, Thomas 74 and Margaret 68 Cowen. James is probably at sea.
Brian Lawson's birth index has Alfred Cowin Maddrell born Douglas 20 Mar 1886 to William Maddrell and Charlotte Jelly.

Joseph is less conclusive.
There is one other possible marriage in the IGI (to Charlotte Kaighin on March 27 1879 in Braddan). However, I did not see this family in any census and cannot identify another possible Joseph.

In 1871 I can find one Joseph as a servant in Lonan aged 22.
In 1881, Rebecca 32 is at 18 Fort Street, Douglas and married with two daughters, Beatrice 8 and Eunice 5, and a Joseph 30 married 30 b Douglas is Master of a vessel in Northumberland, named Maria of Douglas.
In 1891 the only traces that I can find are , Rebecca, who is married aged 41, and with her parents, Thomas and Margaret Kinnish, in Maughold, Eunice 16 a servant in Maughold and Beatrice, a boarder in Liverpool. No Joseph.
Eunice Emily Maddrell married James Thomas Aldis in Maughold on May 5 1895.
Brian's index has a burial record for a 46-year-old Rebecca in Maughold on June 13 1897.
In 1881 there is only one other possible Joseph in the census. He was 37 and was born in Rushen.