First you need to have all documents required to process the residence permit. You might need police clearance which might take time to obtain in UK or Dubai. The best is to find out from the immigration section of the company you will be working for about the documents you require. If you have all the documents, then as soon as you arrive, insist to go for medical immediately unless you are working for an Oil and gas company which does its own medical for its owm staff. Then you will be able to process your residence permit.
The most important document for your child when she is born is birth certificate and passport.Then you can apply for a family Visa for them either on your own or through your company. However, you have to be in Doha to apply for any type of Visa for your family. Alternatively, check if the passports they hold at the time they want to travel with you allows them to get Visa on arrival, then then can come with you without a Visa and will get the Visa on arrival as tourists and then you can convert it to family Visa then the family visa will be converted to residence permit.