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

Eligibility 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.
• * 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.