Phase I/II Dose Escalation and Preliminary Efficacy of CD19 Directed CAR-T Cells Generated Using the Miltenyi CliniMACs Prodigy System (UCD19 CAR-T) in Pediatric Patients With Relapsed and/or Refractory B-Cell Acute Lymphoblastic Leukemia (B-ALL) and B-Cell Non-Hodgkin Lymphoma (B-NHL)

Inclusion Criteria: * Meets clinical criteria for leukapheresis or has a leukapheresis product previously collected and stored per recommended guidelines. * Provision of signed and dated consent form from parent or guardian (patients \<18), the patient themselves (\>18), or legally authorized representative (patient \>18 who lack decision-making capacity); Pediatric patients will be included in age-appropriate discussions and assent will be obtained for those \> 7 years of age, when appropriate, according to institutional standards. * Willingness to participate in long term follow up study. * Stated willingness to comply with all study procedures and be available for the duration of the study. * Males OR non-pregnant, non-breastfeeding females. o Patients of child-bearing potential or capable of fathering a child must agree to use highly effective contraception from the time of initial CAR T cell administration though 12 months following the final administration of investigational product. * Aged 31 days to 30 years (inclusive) at time of consent and enrollment. * Acute Lymphoblastic Leukemia (ALL) OR Non-Hodgkin Lymphoma (NHL) of B-cell origin that: * Has confirmed expression of CD19 by flow cytometry, immunohistochemistry (IHC), or both. Cohort One Criteria: * Meets any one of the following conditions: * Relapsed two or more times * Relapsed at any time after allogeneic BMT * Refractory to standard therapy as determined by the treating physician * Meets criteria for BMT but is ineligible as determined by the treating physician Patient and/or parents declining BMT options and would prefer CAR-T Therapy. * Non-Hodgkin Lymphoma includes all of the following: * Diffuse large B-cell lymphoma (DLBCL) * Burkitt Lymphoma * Intermediate lymphoma between Burkitt and DLBCL * Primary Mediastinal B-cell Lymphoma (PMBL) * Follicular lymphoma * High grade B cell lymphoma * Transformed lymphoma Cohort Two Criteria: * B-ALL in first relapse with any one of the following conditions: * High-risk genomic alterations at initial diagnosis such as KMT2A gene rearrangement, t(17;19), hypodiploidy, Ph-like mutations, BCR-ABL1 fusion (Ph+ ALL), iAMP21, and TP53 inactivating mutation/deletion. * Isolated CNS relapse such that cranial radiation would be indicated as a component of standard salvage therapy. * Down syndrome. * Minimal residual disease (MRD) positivity of \> 0.01% by FACS or \> 0 clonal sequences by NGS in bone marrow post re-induction chemotherapy. * Age 18 years or older. OR Newly diagnosed with persistent MRD ≥ 0.01% by flow cytometry in bone marrow at end of consolidation. * Performance score (Lansky or Karnofsky) of 50% or better; * Unable to or declined to receive commercially available CD19 CAR-T Therapy. Exclusion Criteria: * Evidence of rapidly progressive disease without adequate salvage/bridging regimens as determined by the investigator. * Active Graft-versus-Host Disease (GvHD). * Active, uncontrolled, life-threatening infection that at the determination of the treating physician would preclude safe leukapheresis or tolerance of LD chemotherapy, cell infusion, or cytokine release syndrome. * Evidence of severe organ dysfunction as defined by: * Myocardial dysfunction: Ejection fraction ≤ 40% or shortening fraction ≤ 28%, evidence of physiologically significant pericardial effusion as determined by an echocardiogram (ECHO), and clinically significant electrocardiogram (ECG) findings. * Baseline oxygen saturation of ≤ 90% on room air * Transaminases \> 10x upper limit of normal (ULN) or bilirubin \>2x the ULN, unless thought to be related to primary disease * Estimated Cr clearance \<60 mL/min/1.73 m2 (if nuclear medicine GFR or other more specific testing exceeds this level than it can supersede the estimated clearance) * Post-pubertal females that are pregnant, planning to become pregnant, or unwilling to use birth control (includes abstinence) for the study duration. * Known HIV infection, or active Hepatitis B or active Hepatitis C infection.