You can use a hand wielding a shield emblazoned with a focus to cast somatic components if and only if the spell also requires material components.
Excerpting the relevant sections from the sources linked, we have:
If the spell, such as aid, also has a somatic component, [a holy symbol-emblazoned shield-wielding cleric] can perform that component with the shield hand and keep holding the mace in the other.
If the same cleric casts cure wounds, she needs to put the mace or the shield away, because that spell doesn’t have a material component but does have a somatic component.
A hand with a material component, including a spellcasting focus, can perform somatic components
A holy symbol can function as a spellcasting focus even when it's on a shield.
Crawford's tweets don't dissent from the information given in the Sage Advice column (for once), remaining consistent. Shield w/ Holy Symbol == (non-consumed, no-cost) material component, material component + somatic component can always be done together regardless of what that material component is, and a hand with a shield cannot use a somatic component unless it is also fulfilling the material component.
In addition, in RAW only divine casters can "emblazon." So Warlocks are automatically left out.
Even if a DM allowed an arcane caster to emblazon, though, Eldritch Blast does not have a material component. Consequently, without some feat or circumstance that changes the casting rules, it cannot be performed by a Warlock with a mace or rod in one hand and a shield in the other.