This phase II trial studies how well giving an umbilical cord blood transplant together with cyclophosphamide, fludarabine, and total-body irradiation (TBI) works in treating patients with hematologic diseases. Giving chemotherapy, such as cyclophosphamide, fludarabine and thiotepa, and TBI before a donor cord blood transplant (CBT) helps stop the growth of cancer and abnormal cells and helps stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells. Giving cyclosporine and mycophenolate mofetil after transplant may stop this from happening in patients with high-risk hematologic diseases.

OUTLINE: Patients are assigned to 1 of 2 arms.

ARM I: Patients aged 6 months through 30 years old receive myeloablative conditioning comprising fludarabine intravenously (IV) over 30 minutes on days -8 to -6, cyclophosphamide IV on days -7 and -6, and undergo high-dose TBI twice daily (BID) on days -4 to -1. Patients then undergo UCBT on day 0. Patients undergo blood sample collection throughout the study. Patients undergo echocardiography (ECHO) or multigated acquisition scan (MUGA) and diagnostic imaging during screening and as clinically indicated on study. Patients also undergo blood sample collection throughout the study and bone marrow aspirate during screening and on study.

ARM II: Patients aged 6 months through 65 years old receive myeloablative conditioning comprising fludarabine IV over 30-60 minutes on days -6 to -2, cyclophosphamide IV on day -6, thiotepa IV over 2-4 hours on days -5 and -4, and middle-intensity TBI once daily (QD) on days -2 and -1. Patients undergo ECHO or MUGA and diagnostic imaging during screening and as clinically indicated on study. Patients also undergo blood sample collection throughout the study and bone marrow aspirate during screening and on study.

All patients receive GVHD prophylaxis comprising cyclosporine IV over 1 hour every 8 or 12 hours, then cyclosporine orally (PO) (if tolerated), on days -3 to 100 with taper on day 101. Patients also receive mycophenolate mofetil IV every 8 hours on days 0 to 7 and then PO (if tolerated) three times daily (TID) on days 8-30. Mycophenolate mofetil is tapered to BID on day 30 or 7 days after engraftment if there is no acute GVHD, and then tapered over 2-3 weeks beginning on day 45 (or 15 days after engraftment if engraftment occurred > day 30) after engraftment if there continues to be no evidence of acute GVHD.

After completion of study treatment, patients are followed up at day 180, 1 year, and 2 years.

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.