Cord Blood Transplant With Dilanubicel for the Treatment of HIV Positive Hematologic Cancers

Inclusion Criteria: * \>= 6 months to =\< 65 years * Treatment with combination antiretroviral therapy (cART) for at least 1 month before enrollment * Viral load \< 5000 copies/ml plasma on cART * Disease criteria * Acute myeloid leukemia * High risk in first complete remission (CR1), \>= 2 cycles to obtain complete remission (CR), erythroblastic or megakaryocytic leukemia; \>= in second complete remission (CR2) * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15% * Patients for whom adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible. Specimen for morphologic assessment, including possible repeat procedures will be obtained (as possible). These patients must be discussed with the lead principal investigator, Filippo Milano prior to enrollment * Acute lymphoblastic leukemia * High risk CR1 (for example, but not limited to: t(9;22), t(1;19), t(4;11) or other mixed-lineage leukemia \[MLL\] rearrangements, hypodiploid); greater than 1 cycle to obtain CR; \>= CR2 * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15% * Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible. Specimen for morphologic assessment, including possible repeat procedures will be obtained (as possible). These patients must be discussed with the lead principal investigator, Filippo Milano prior to enrollment * Chronic myelogenous leukemia excluding refractory blast crisis. To be eligible in first chronic phase (CP1) patient must have failed or be intolerant to imatinib mesylate * Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate (Int)-2 or high risk (i.e., refractory anemia with excess blasts \[RAEB\], refractory anemia with excess blasts in transformation \[RAEBt\]) or refractory anemia with severe pancytopenia or high-risk cytogenetics. Blasts must be \< 10% by a representative bone marrow aspirate morphology * Other hematologic malignancy such as non-Hodgkin lymphomas. Fred Hutch site: These patients must be presented at Patient Care Conference (PCC) prior to enrollment, given potential competing eligibility on auto-transplant protocols. Participating centers: These patients must be discussed with the lead principal investigator, Filippo Milano prior to enrollment * Karnofsky (\>= 16 years old) \>= 70% * Lansky (\< 16 years old) \>= 50% * Adults: Calculated creatinine clearance must be \> 60 mL and serum creatinine =\< 2 mg/dL * Children (\< 18 years old): Calculated creatinine clearance must be \> 60 mL/min * Total serum bilirubin must be \< 3 mg/dL * Transaminases must be \< 3 x the upper limit of normal * Diffusion capacity of the lung for carbon monoxide (DLCO) corrected \> 50% normal or for pediatric patients in whom DLCO cannot be measured has adequate pulmonary function * Left ventricular ejection fraction \> 45% OR * Shortening fraction \> 26% * Ability to understand and the willingness to sign a written informed consent document (adult subject or parent/legal guardian of minor subject) Exclusion Criteria: * Uncontrolled viral or bacterial infection at the time of study enrollment * Active or recent (prior 6 month) invasive fungal infection without infectious disease (ID) consult and approval * Pregnant or breastfeeding * Prior myeloablative transplant within the last 6 months * Extensive prior therapy including \> 12 months alkylator therapy or \> 6 months alkylator therapy with extensive radiation * Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy. Diagnostic lumbar puncture to be performed