TRACON Controller reported an aircraft was below MVA but initially went unnoticed while the controller managed a NORDO aircraft.

Narrative

Aircraft X had departed ZZZ. I had to step them up reference other ZZZ traffic behind. Initially I gave 120; then 140. I told the jet behind them to climb to 230 and expedite through 170 because I planned on giving 160 to Aircraft X as that is the MVA eventually. At some point I did give him 150 only because the aircraft behind him was still not climbing super great.During this time an aircraft on approach to ZZZ1 went nordo; I had to try and reach out to them at least 3 times. Then a departure off of ZZZ1 kept calling; waiting to be radar identified. And a departure off of ZZZ2 was flying towards ZZZ1 at 8 and I had to figure out when I could climb them reference ZZZ and ZZZ1 traffic. Eventually the nordo responded; and at that time I noticed the CLIMB red letters on Aircraft X; alerting me that he was still at 150 approaching/in the 160 MVA.I forget exactly what order I said it in; but I know I didn't say Low Altitude Alert first; I just wanted him to start climbing; so I told him to start climbing; then issued the LA alert phraseology. I also asked if he had terrain in sight and he said negative; he was IMC. I apologized and had him climb. Recommendation: They do already split the position into XX/XY often; and that would have helped because the nordo and departure off ZZZ1 would not have taken my attention off of the climb for Aircraft X. So it is something we already do but usually around the 11 o'clock push. Maybe when it is IMC weather like this and we know all the little guys come out and clog up the frequency; it should be split. Also; putting it into the Letter of Agreement (LOA) that ZZZ *has* to give us 5-7 miles for a jet behind a prop would be beneficial. We would still have to tell the back traffic to expedite climb through 180 probably depending on type; but maybe not as imminent. From my understanding it is in the Tower's SOP as a best practice; but an actual rule in the LOA would be nice.