This phase II trial studies how well an umbilical cord blood transplant with added sugar works with chemotherapy and radiation therapy in treating patients with leukemia or lymphoma. Giving chemotherapy and total-body irradiation before a donor umbilical cord blood transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer 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. The umbilical cord blood cells will be grown ("expanded") on a special layer of cells collected from the bone marrow of healthy volunteers in a laboratory. A type of sugar will also be added to the cells in the laboratory that may help the transplant to "take" faster.

PRIMARY OBJECTIVES:

I. To evaluate the safety and feasibility of transplantation of cord blood which is expanded in mesenchymal precursor cell (MPC) co-cultures then fucosylated with fucosyltransferase (FT)-VI and guanosine diphosphate (GDP) fucose prior to infusion in patients with hematologic malignancies following high-dose therapy.

II. To evaluate the time to engraftment using expanded fucosylated cord blood.

SECONDARY OBJECTIVES:

I. To evaluate the rate and severity of graft versus host disease. II. To evaluate the rates of infectious complications. III. To evaluate the rates of disease-free and overall survival.

Summary: All patients receive Busulfan as per standard of care. Busulfan test dose can be administered either as an outpatient prior to admission or as an inpatient on Day -10.

Busulfan pharmacokinetics will be performed with the test dose and the first dose on Day -7 per standard of care. The doses of Days -6, -5, and -4 will be subsequently adjusted to target an AUC of 4,000 microMol.min-1. In the event that PK adjusting were not possible a dose of busulfan of 130 mg/m2 will be administered.

GVHD PROPHYLAXIS: All patients receive mycofenolate mofetil IV over 2 hours or orally (PO) twice daily (BID) on days -3 with a taper beginning on day 100 in the absence of GVHD, tacrolimus IV or PO starting on day -2 for 6 months in the absence of GVHD, and filgrastim-sndz subcutaneously (SC) once daily (QD) starting on day 0 until white blood count begins to recover.

After completion of study treatment, patients are followed up at months 1, 3, 6, and 12.

Inclusion Criteria:

• Patients must have one of the following hematologic malignancies: a. Acute myelogenous leukemia (AML), induction failure, high-risk for relapse first remission (with intermediate-risk or high-risk cytogenetics, flt3 mutation positive and/or evidence of minimal residual disease by flow cytometry), secondary leukemia from prior chemotherapy and/or arising from myelodysplastic syndrome (MDS), Langerhan's cell histiocytosis, any disease beyond first remission. b. Myelodysplastic syndrome (MDS): MDS International Prognostic Scoring System (IPSS) INT-1 will be enrolled only if the subjects have failed previous leukemia treatments and are transfusion-dependent. MDS may be primary or therapy related, including patients that will be considered for transplant. Including the following categories: 1) Revised IPSS intermediate and high risk groups, 2) MDS with transfusion dependency, 3) Failure to respond or progression of disease on hypomethylating agents, 4) Refractory anemia with excess of blasts, 5) Transformation to acute leukemia, 6) Chronic myelomonocytic leukemia, 7) Atypical MDS/myeloproliferative syndromes, 8) Complex karyotype, abn(3g), -5/5g-, -7/7g-, abn(12p), abn(17p). c. Acute lymphoblastic leukemia (ALL) patients with the following will be considered: induction failure, primary refractory to treatment (do not achieve complete remission after first course of therapy) or are beyond first remission including second or greater remission or active disease. Patients in first remission are eligible if they are considered high risk, defined as any of the following detected at any time: with translocations 9;22 or 4;11, hypodiploidy, complex karyotype, secondary leukemia developing after cytotoxic drug exposure, and/or evidence of minimal residual disease, or acute biphenotypic leukemia which excludes > 7 chromosomal abnormalities, or double hit non-Hodgkin's lymphoma. Non-Hodgkin's lymphoma (NHL) in second or third complete remission or relapse (including relapse post autologous hematopoietic stem cell transplant), or relapsed double hit lymphoma. Small lymphocytic lymphoma (SLL), or chronic lymphocytic leukemia (CLL) with progressive disease with progression after standard of care therapy or have failed/been intolerant to ibrutinib. Chronic myelogenous leukemia (CML) second chronic phase or accelerated phase. Hodgkin's disease (HD): Induction failure after the first complete remission, or relapse (including relapse post autologous hematopoietic stem cell transplant), or those with active disease.
• The first 6 patients must be >= 18 and =< 65 years old. The subsequent patients may include pediatric patients >= 12 and =< 65 years old. Eligibility for pediatric patients will be determined in conjunction with an MD Anderson Cancer Center (MDACC) pediatrician.
• Performance score of at least 80% by Karnofsky or performance score (PS) < 3 (Eastern Cooperative Oncology Group [ECOG]) (age >= 12 years)
• Left ventricular ejection fraction of > 40%.
• Pulmonary function test (PFT) demonstrating a diffusion capacity of least 50% predicted.
• Creatinine =< 1.5 mg/dL for patients 12 years old and older and =< 1 for patients younger than 12 years old.
• Serum glutamate pyruvate transaminase (SGPT) =< to 2.0 x normal.
• Bilirubin =< to 2.0 x normal.
• Negative beta human chorionic gonadotropin (HCG) test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization and willing to use an effective contraceptive measure while on study.
• Patients must have two cord blood (CB) units available which are matched with the patient at 4, 5, or 6/6 HLA class I (serological) and II (molecular) antigens. Each cord must contain at least 1.5 x 10^7 total nucleated cells/Kg recipient body weight (pre-thaw).
• Have identified a backup cells source in case of engraftment failure. The source can be autologous, related or unrelated.
• Patient must not have a 10/10 HLA matched family member or unrelated donor.
• Patients will have a back-up graft from any of the following: an available fraction of autologous marrow; or peripheral blood progenitor cells (PBPCs) harvested and cryopreserved; or family member donor; or a third cord blood unit.
• Prior to initiating chemotherapy in this study, twenty-one or more days must have elapsed since the patient's last radiation or chemotherapy administration (Hydrea, Gleevec and other tyrosine kinase inhibitors [TKI] as well as intrathecal therapy are accepted exceptions).

Exclusion Criteria:

• Patients with known history of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS).
• Patients with positive hepatitis serology that is definitive of active disease.
• Active central nervous system (CNS) disease in patient with history of CNS malignancy.
• Patients with chronic active hepatitis or cirrhosis. If positive hepatitis serology, the study chair may deem the patient eligible based on the results of liver biopsy.
• Patients with uncontrolled serious medical condition such as persistent septicemia despite adequate antibiotic therapy, decompensated congestive heart failure despite cardiac medications or pulmonary insufficiency requiring intubation (excluding primary disease for which CB transplantation is proposed), or psychiatric condition that would limit informed consent.
• Positive beta HCG in female of child-bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization or breast-feeding.
• Pediatric patients with acute lymphoblastic leukemia (ALL) that is t (9,22) positive in first remission are not eligible unless there is evidence of minimal residual disease after initial induction and/or consolidation treatment or the pediatric Philadelphia chromosome positive (Ph+) ALL is clinically refractory to available therapies with evidence of persistence in the bone marrow or peripheral blood.
• Patients with options for treatment that are known to be curative are not eligible.