Do the English have subjunctive mood?
Yes, but it is subtle.
The subjunctive mood conveys things that are not facts. This includes beliefs, wishes, and intentions.
English's subjunctive is hard to spot--in fact, many native speakers do not properly conjugate the subjunctive and do not know when they're using it.
As an example, here is (almost) the same sentence conveyed in the indicative mood, and then the subjunctive mood:
Indicative: I wished to attend the party.
Subjunctive: He wished that I attend the party.
The subjunctive mood was used to convey his wish. However, even though the mood was employed, there was no change in the actual formation of the verb--the word "attend" was used in both cases.
The rule for forming the present subjunctive is that the infinitive form of the verb is always used, and the phrase always uses the word "that."
Indicative: He is at his wife's side at the hospital.
Subjunctive: It is imperative that he be at his wife's side at the hospital.
Notice that "he is" in the indicative mood shifts to "that he be" in the subjunctive mood.