Assuming I made a bracha before davening and the drink is something that is muttar to drink before davening, are there any points from the time I put on my tefillin to the end of davening that it’s permissible to drink on a weekday?
From a technical aspect, I am not aware of any halacha that says that we may not drink during davening. As far as eating or drinking while one is wearing tefillin, the M:B 40-19 brings an opinion that questions is it is permitted or not. However, others say that if one is going to be wearing his tefillin for a while that it is permitted, but if he will be taking them off soon, that he should abstain from eating or drinking. (See Piskei Teshuvos 40-8) .
As a side point, if someone needs a drink during davening, (a bracha was made beforehand) going outside and taking a quick drink is one thing, but to bring a cup of coffee into shul to sip while one is davening is wholly disrespectful to the davening. A person that will be going to an executive’s office for an important meeting, i.e. you need a big loan from the executive, you wouldn’t dream of bringing a coffee along with you to sip while you humbly ask the executive for the loan. When we are davening, we are given the tremendous opportunity to talk to Hashem our king, praise Him and ask Him for our life and needs. Davening is an exercise of humility, therefore bringing along a drink is very much out of place, and contrary to the while idea of what we are doing when we are davening.