I came across this while searching for something else. I’m fairly new to going through the immigration process myself, but pretty familiar with the process due to friends and family and professional reasons.
The posters who more or less said your use of a completely legal method shouldn’t be allowed should keep to commenting on the actual question raised. Your case is actually a poster child for why Congress permits AOS from ESTA in certain circumstances — humanitarian reasons. Circumstances change and WT admission class is default 90 days. It would be inhumane for the government to require family members visiting and assisting with legitimate family needs to go back to their home country when there might be a family emergency, change of medical circumstances, change of economic circumstances, etc. If anything, in contrast to what was harshly said to you, your case highlights why Congress should act to streamline family reunification visas via consular processing, not why it should repeal good public policy.
The caregiver point @EmilyW raised is debatable — if your mother is with you and helping you out without it raising to the level of a full time job, that’s arguably within the intent of B-2. Even if it wasn’t, B-1 permits domestic workers in limited circumstances (business people temporarily stationed in the U.S., different circumstances, but similar principles if you go through the FAM guidance — anticipated limited time working on entry, and motivation is that there’s humanitarian hardship telling a family they can’t bring in their foreign nanny for 2 months, etc.)
FAM is limited to visa issuance but the point on permitting domestic workers of businessmen shows the guidance around what is/isn’t work on a B visa isn’t as clear cut as people think — it’s not simply 'could an American be paid for this.'
From a purely practical perspective, I don’t think USCIS is going to come down hard on a British grandmother visiting her family and helping care for a grandchild. It’s an overly expansive reading of law/guidance without realizing that there is a grey area on what’s permissible under B visas that’s common in immigration internet sites. The practical reality is USCIS is aware that there’s multiple levels of hearings and appeals they’d have to go through after a denial for your mother to be forced to leave the country and that they’d likely lose at some point along the way. That has an impact on any decision.
Finally, the advice that @OldUser gave on the process to follow at the beginning is the answer to your actual question. I’m so sorry your thread got derailed after someone gave you the correct answer.
Advice on forums like this helped a lot of people I know and it’s why I’m here too, but its also worth remembering that based on both the TOS and the culture in a lot of online communities, people tend to take a stricter view of the law than the government does on some things. Helps when people are starting from scratch for them to avoid the grey areas, but once someone is already there, it only makes people more stressed out in an already stressful process.