Hi, there are several hospital or clinics that will surely help you and your child. Most hospital are located in Mataram such as Rumah Sakit Umum, Risa Hospital in Cakranegara (near Mataram mall), and RS Harapan Keluarga in Selagalas east of the city. Here are their details.
Medical Clinics
Holiday Resort Lombok-Medical clinic
Located at Mangsit beach, Senggigi (halfway through Mangsit beach,the clinic is downstairs, on northeast corner of main building).
This clinic has qualified nurse and doctor on call.
Phone: +62 370 693 444
Fax: +62 370 693 092, +62 370 693 206
Klinik Risa (Risa Centra Medika Hospital)
Located at Jalan Pejanggik 115, Cakranegara (just east of Mataram Mall on the right hand side of the road). The clinic has a 24 hours emergency room. Full hospital facilities are available, ie: consulting rooms and dentist.
Phone: +62 370 625 560
Puskesmas Senggigi (Local Health Clinic Senggigi Township)
Located at Jalan Raya Kerandangan (Situated on the main road at the southern end of Kerandangan just past Puri Saron and Pacific Hotel). Very limited hours, normally in the evenings. This is a local municipal health clinic catering to the residents of the surrounding district rather than an emergency medical centre.
Emergency Services
Lombok Regional Police (Polda)
Located at Jalan Gajah Mada 7, Ampenan,
Phone: +62 370 693 110, 110 (24 hours emergency number)
Tourism Police Senggigi (Pos Polsek Senggigi)
Located at Jalan Raya Senggigi km 1, Senggigi (near the Art Market)
Phone: +62 370 632 733
Ambulance
In emergency 118. The reality is that unless within the confines of the main city of Mataram/Ampenan/Cakranegara an ambulance is normally too far away to provide prompt transport in the case of a true emergency. Most often a taxi, police vehicle or a private car is used to get someone requiring urgent medical attention to a hospital. The ambulances in Lombok are more often used in the role of assisted patient transport rather than for first responder/paramedic supported emergency assistance and transport.
Phone: +62 370 623 489
As for your next destination, I suggest you go to Gili Trawangan, near Lombok. It has beautiful beaches and definitely a perfect place to rest.
For nappies, and baby stuff, I suggest you buy it from Denpasar, Bali since its cheaper. Or you can buy on Mataram local shop.