Cord Blood Transplant, Cyclophosphamide, Fludarabine, and Total-Body Irradiation in Treating Patients With High-Risk Hematologic Diseases

Inclusion Criteria: * Patients aged 6 months to =\< 65 years at time of consent. * Acute myelogenous leukemia (AML): * Complete first remission (CR1), complete second remission (CR2) or greater (CR2+), must have \< 5% marrow blasts at the time of transplant. * Patients in morphologic remission with persistent cytogenetic, flow cytometric, or molecular aberrations are eligible. * Acute lymphoblastic leukemia (ALL): * Complete first remission (CR1) at high risk for relapse such as any of the following: * Presence of any high-risk cytogenetic abnormalities such as t(9;22), t(1;19), t(4;11) or other MLL rearrangements (11q23) or other high-risk molecular abnormality. * Failure to achieve MRD- complete remission after induction therapy. * Persistence or recurrence of minimal residual disease on therapy. * Any patient unable to tolerate consolidation and/or maintenance chemotherapy as would have been deemed appropriate by the treating physician. * Other high-risk features not defined above. * Complete second remission (CR2) or greater (CR2+). * Note: ALL with less than 5% blasts at time of transplant but persistent cytogenetic, flow cytometric or molecular aberrations are eligible. * Other acute leukemias: Acute leukemias of ambiguous lineage or mixed phenotype with less than 5% blasts. Leukemias in morphologic remission with persistent cytogenetic, flow cytometric or molecular aberrations are eligible. * Chronic Myeloid Leukemia (CML): Excluding refractory blast crisis. To be eligible in first chronic phase (CP1) patient must have failed or be intolerant to tyrosine kinase inhibitor therapy. * Myelodysplastic syndromes (MDS) and myeloproliferative disorders (MPD) other than myelofibrosis: * MDS/MPD overlap syndromes without myelofibrosis. * MDS/ MPD patients must have less than 10% bone marrow myeloblasts and absolute neutrophil count (ANC) \> 0.2 (growth factor supported if necessary) at transplant work-up. * Non-Hodgkin lymphoma (NHL) at high-risk of relapse or progression if not in remission: * Eligible patients with aggressive histology (such as, but not limited to, diffuse large B-cell NHL, mantle cell NHL, and T-cell histology) in CR by PET/CT imaging. * Eligible patients with indolent B-cell NHL (such as, but not limited to, follicular, small cell or marginal zone NHL) will have 2nd or subsequent progression with PR or CR by PET/CT imaging. * Blastic plasmacytoid dendritic cell neoplasm (BPDCN) in morphologic remission. * Only for adult patients, to prevent graft rejection, patients who received only non-lymphodepleting agents for their malignancy (hypomethylating agents, venetoclax, hydroxyurea, TKIs etc.), or patients who received lymphodepleting chemotherapy \> 3 months prior to scheduled admission, may receive fludarabine 25 mg/m\^2 daily x 3 days for lymphodepletion 14-42 days (aiming for 2-4 weeks) at the discretion of the principal investigator (PI). * For patients \> 18 years old, Karnofsky score ≥ 70%. For patients =\< 18 years old, Lansky score ≥ 50%. * Calculated creatinine clearance \> 70 ml/min. * Bilirubin \< 1.5 mg/dL (unless benign congenital hyperbilirubinemia or hemolysis). * Alanine transaminase (ALT) \< 3 x upper limit of normal (ULN). * For patients \> 18 years old, pulmonary function (spirometry and corrected diffusing capacity for carbon monoxide \[DLCO\]) \> 60% predicted. For patients =\< 18 years old, or any patient unable to perform pulmonary function tests, O2 saturation \> 92% on room air. * Left ventricular ejection fraction \> 50%. * Albumin \> 3.0 g/dL. * For patients \> 18 years old, Hematopoietic Cell Transplantation Comorbidity index (HCT-CI) =\< 5. * UCB units will be selected according to current umbilical cord blood graft selection algorithm. One or two UCB units may be used to achieve the required cell dose. * The UCB graft is matched at 4-6 HLA-A, B, DRB1 antigens with the recipient. This may include 0-2 antigen mismatches at the A or B or DRB1 loci. Unit selection based on cryopreserved nucleated cell dose and HLA-A, B, DRB1 using intermediate resolution A, B antigen and DRB1 allele typing. Exclusion Criteria: * Diagnosis of myelofibrosis or other malignancy with moderate-severe bone marrow fibrosis. * Patients persistent with central nervous system (CNS) involvement in cerebrospinal fluid (CSF) or CNS imaging at time of screening0 * Prior checkpoint inhibitors/ blockade in the last 12 months. * Two prior stem cell transplants of any kind. * One prior autologous stem cell transplant within the preceding 12 months. * Prior allogeneic transplantation. * Prior involved field radiation therapy that would preclude safe delivery of 400cGy total body irradiation (TBI) in the opinion of radiation oncology. * Active and uncontrolled infection at time of transplantation. * HIV infection. * Inadequate performance status/ organ function. * Pregnancy or breast feeding. * Patient or guardian unable to give informed consent or unable to comply with the treatment protocol including appropriate supportive care, long-term follow-up, and research tests.