Yes, in the form of an ape, you can.
The only reason why the druid of lower then 18 level can't cast spells with no verbal component while shaped into an ape, is Wild Shape feature specifically stating, that he can't. He can use either focus, pouch or individual components as any creature with hands can.<br>
When druid gains Beast Spells, he no longer restricted to intelligible speech or hands to use verbal or somatic component. But his ability to use any components his form capable of using stays with him. He won't suddenly forget how to hold his staff-like focus for example.<br>
Despite slightly obscure wording, Beast Spells don't spoil any of your abilities that you already have. Beast Spells just says it can't grant you the ability to use material component if you can't already use it (opposed to verbal and somatic ones which you now can use despite you usually can't without that class feature).
I may also add that you can choose to "wear" your focus instead of dropping it to the ground and then picking it up. An ape is capable of wearing (i.e. being equipped with) a club-like thing that your focus is.