First I'll try to make things clear.
Wali: Is the guardian of a lady. If alive the 1st and only choice is her father and after that the closest male relatives (but only if the father doesn't exist) such as brother or grandfather (the correct order of which is to be asked first needs to be checked as there are different opinions according madhhabs, see Who can be wali, and in what order?).
The woman’s guardian is her father, then his father (her grandfather), then her son, then his son (her grandson – if she has a son), then her full brother, then her brother through her father only, then her brothers’ sons, then paternal uncles, then their sons, then the father’s paternal uncles, then the ruler. See: al-Mughni (9/355).
Wakil/Wakeel: Is any person who is legally appointed by the Wali to represent him in a situation like marriage: In this case he is basically ordered to say whether the wali is accepting the marriage or refusing it.
This means either the father is needed to be present during the 'aqd an-Nikah or he needs to be represented by his wakeel!
So if the option of a wakeel is available the father just need to appoint him by a letter in which he (if necessary) even make his "acceptance or refutation" clear and the most important content of this letter is that this wakeel is clearly appointed by the father (for example by an IDCard# or anything that would help to clearly identify him).
Else if the father refuses for no good reasons ((For example the boy is not a practicing Muslim or drinks alcohol etc.) the girl can go to a shari'a court and ask them to take the guardianship away or repeal it from her father.
Note that however this might be a legal and permissible option it would be a time-killing act especially if the judge is not willing or accepting to do so, except if father is really out of his mind, for any reason and don't want to let his daughter get married at all.
To be honest an Imam or judge can be a guardian if the father is not present, but no Imam or judge who knows the rulings of Allah well would accept this duty if the guardian (father) of the (born Muslim) girl is alive! A clear exception is the case of a convert girl, as non-Muslims can't be guardian for a Muslim woman (but in case of marriage among women from people of the book the rule of guardianship applies), here it's usual for an Imam or Judge to play the role of the guardian.
So if possible the girl should convince her father to appoint a wakeel and convince him to accept the marriage. If necessary with the help of her future husband or any relative (like her auntie) who could help her to convince her guardian.
Note that anything anybody except the father is saying is irrelevant. As only the father can accept or refuse the marriage as a guardian, so in case he is delaying the marriage due to respect for his family he might be committing a sin and therefore the intervention of a court is highly requested and recommended. As the father doesn't fulfill his duty as guardian in the correct manner. But maybe things can be spoken out before taking this step!
This fatwa on islamqa #247217 indirectly gives a solution:
If a woman gets married without a guardian, or the imam of the mosque does her marriage contract even though she has a guardian, then there is a difference of opinion among the fuqaha’ concerning such a marriage. If it happens, it is not to be annulled, in light of the difference of scholarly opinion concerning that.
so maybe getting married by the Imam of a mosque could be a solution, but you must check or get certain that this marriage would not be annulled by the authorities at where you live... This could fall under trickery, but might be a solution.
In any case I would recommend the girl to think and reflect about it long and deep enough as marriage is a decision for life or at least should be! Both parties must be aware that this solution might cause big issues in the relationship between the spouses and the family of the wife!
The fatwa also describes the legal way of "transferring" the guardianship to an other guardian.