Another option that might seem less forceful is encourage, as in:
I would really appreciate it if I could get the results earlier. Could you please help me encourage them to give me the results?
(Other edits in italics)
From the Oxford Living Dictionaries:
encourage
VERB
[WITH OBJECT]
Give support, confidence, or hope to (someone)
‘we were encouraged by the success of this venture’
1.1 [with object and infinitive] Persuade (someone) to do or continue to do something by giving support and advice.
‘pupils are encouraged to be creative’
1.2 Stimulate the development of (an activity, state, or belief)
‘the intention is to encourage new writing talent’
Note especially definitions 1.1 and 1.2.