• History or status of a histologically-confirmed hematological malignancy that is expected to express CD19 and CD20; relapse after or failure to respond to at least one prior treatment regimen; no available treatment options that are expected to prolong survival (Part I and II); relapsed after or failed to respond to only one prior systemic treatment regimen (Part III)

• Must have at least one measurable target lesion (\>/= 1.5 cm) in its largest dimension by computed tomography scan

• Able and willing to provide a fresh biopsy from a safely accessible site, per Investigator's determination, providing the participant has more than one measurable target lesion

• Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1, or \</= 2 for some participants in Part III

• Life expectancy of \>/= 12 weeks

• Adverse events from prior anti-cancer therapy must have resolved to Grade \</= 1

• Adequate liver, hematological, and renal function

• Negative test results for acute or chronic hepatitis B virus infection

• Negative test results for hepatitis C virus and HIV

• The contraception and abstinence requirements are intended to prevent exposure of an embryo to the study treatment. The reliability of sexual abstinence for male and/or female enrollment eligibility needs to be evaluated in relation to the duration of the clinical study and the preferred and usual lifestyle of the participant. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post-ovulation methods) and withdrawal are not acceptable methods of preventing drug exposure

• Female participants: A female participant is eligible to participate if she is not pregnant and not breastfeeding, and if at least one of the following applies: women of non-childbearing potential (WONCBP); women of child bearing potential (WOCBP) who agree to remain abstinent or use two highly effective contraceptive methods with a failure rate of \<1% per year during the treatment period and for at least 18 months after obinutuzumab or 3.5 months after the last dose of englumafusp alfa, 2 months after last dose of glofitamab, or 3 months after the last dose of tocilizumab, whichever is longer. Examples of contraceptive methods with a failure rate of \< 1% per year include bilateral tubal occlusion/ ligation, male sexual partner who is sterilized, established proper use of hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices and copper intrauterine devices. Hormonal contraceptive methods must be supplemented by a barrier method; have a negative pregnancy test (blood) within the 7 days prior to the first study treatment administration

• Male participants: During the treatment period and for at least 3 months after obinutuzumab, or 3.5 months after the last dose of englumafusp alfa, 2 months after the last dose of glofitamab, or 2 months after the last dose of tocilizumab whichever is longer, agreement to: Remain abstinent or use contraceptive measures such as a condom plus an additional contraceptive method that together result in a failure rate of \< 1% per year, with a partner who is a women of childbearing potential. With pregnant female partner, remain abstinent or use contraceptive measures such as a condom to avoid exposing the embryo; refrain from donating sperm during this same period