You've got one bit backward: You can apply for a visa up to three months before travel. It is the earliest you can apply, not the latest.
As for obtaining a visa quickly, you can pay for priority visa service for a decision within 5 working days or super priority service for a decision within 24 hours when you apply for the visa.
What you're looking for is an aide or assistant. That is employment. Your entire venture will fail if you do not understand that fact.
Look into Part 8 (family members) or Part 5 (domestic workers in private households), note that "raising a daughter" counts. Note also 159G in part 5.
"But I aim to pay them nothing" - that makes no difference in ...
This UK govt page notes that the fee charged depends on visa duration. The text is:
A Standard Visitor visa costs £95.
The fee for a long-term Standard Visitor visa depends on its length:
2 years - £361
5 years - £655
10 years - £822
When you get to the "payments" page, you'll have to choose the amount of the payment, and ...
Get the UK visa first.
A Schengen Visa can only be applied for 3 months before your trip
assume 1st April (earliest then 2nd of January)
Your are 'getting' your UK visa in February
when will that application be made (during which you may not have your passport)?
Since you are moving to the UK, that should be your priority
which will also make ...
When you apply for a visa, it really helps if your destination country believes that you will leave again. For a Japanese going to the US, or a German going to Turkey, that is not much of an issue. Relatively few Japanese are trying to overstay in the US. For a Pakistani going to the UK, the worry is greater. Other examples fall in between.
If you were an ...