This could have been us writing this as it is very simular to my mums story. The difference was we didnt realise what was going on when she got lost etc. Then I found her uncontious and she was taken to hospital. Four weeks later she had tests and the consultant told us she had Alzheimers and because she was a wonderer from the hospital they didnt think she could live on her own again. We did try and get help and like you and my two children we all work and would have to give up careers to look after her and that would mean loosing my house. You need to find a home yourself. One that you are happy with. Not a nursing home as she is not physically ill but a residental home with a dementia unit. Meanwhile work on social services to get her into a home or your choice. Check it out and make sure that the care is good. Also get her to sign a Power of Attorney because the onset could come on very quickly and you will not be able to handle any of her affairs, no matter how small or great they may be.
I would say let her go missing but follow her and when she cannot find her way home ring social services emergency number and say that she is missing.
Let them look for her for a few hours. Then you say you have found her but she is very confused and you need help to get her home otherwise you will need to ring the police or an ambulance to help.
When they come and try to place her in a home of their choice you have a home that you want up your sleeve and get her into that one.
You must get her to sign a power of attorney asap. before she is incapable.