In the Denmark Baptisms, there are three records that could match Neils,
Niels Peder Rasmussen, Christened 07 Oct 1875, VEFLINGE, ODENSE, DENMARK, father Rasmus Pederson, Mother Maren Olsen
"Denmark Baptisms, 1618-1923," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XYCJ-LD4 : 6 December 2014), Niels Peder Rasmussen, 07 Oct 1875; citing , VEFLINGE, ODENSE, DENMARK; FHL microfilm 50,489.
Niels Peder Rasmussen, Christened 29 Mar 1875 1875, IDESTRUP, MARIBO, DENMARK, father Lars Resmussen Kaare, Mother Ane Rasmussen
"Denmark Baptisms, 1618-1923," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XY6C-LHG : 6 December 2014), Niels Peder Rasmussen, 29 Mar 1875; citing , IDESTRUP, MARIBO, DENMARK; FHL microfilm 49,648.
Niels Peder Rasmussen, Christened 17 May 1875, VIGERSLEV,ODENSE,DENMARK, birthdate 17 Dec 1874, father Resmus Jensen Hansen, Mother Anne Elisabeth Larsen
"Denmark Baptisms, 1618-1923," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XYKP-KMG : 6 December 2014), Niels Peder Rasmussen, 17 May 1875; citing ; FHL microfilm 312,503.