German visa while in UK for Pakistani Passport Holder

It might possible provided you fulfil certain conditions but its highly discouraged. Apply from your country of permanent residence.

To apply for a Schengen visa from the UK you must be a UK resident for at least three months. If this condition is not fulfilled, the Consulate of your country of preference will hardly take the application into consideration.

After 3 months of your stay you May apply, however

Non-residents such as travelers and tourists who plan on staying in the UK for 3 months or less should be discouraged from applying for a Schengen visa from within the UK. Instead, they should apply in their home country so that the application is processed, chances of getting a positive answer become greater.

Source

Also

Please note that if you are not a resident in the United Kingdom you should apply at the German Embassy in your country of residence.

Source

Lastly, a very authentic source, VFS Global, states the following elibility

Eligibility

Applicant must at least meet the following criteria in order to be eligible to submit a Schengen visa application:

1.holding a valid passport/travel document

2.holding a valid UK residence permit. In case applicant holds a short stay visa, a C type visa (see example here), for the UK, then applicant needs to apply for a Schengen visa from his/her residence country.

3.applying for a short stay visa (Tourist: including Minors, Visitor Family/Friends, Business, EEA/EU and Swiss national family member or Airport Transit)

(The emphasis is mine)

This case does not fulfil eligibility requirement number 2