Well I'm not able to use a mobile phone, I don't have one...but I can use a pump. I also know what its like to babysit grandchildren. Ok I've managed to bring up children myself but there's still that scary feeling you'll do something wrong! Add diabetes and new technology, you're adding a lot of pressure. You're worried, so I'm sure is your Mum.
Seriously, you need to be easily contactable and not far away for the first few times, don't go away for a day before your Mum has got used to dealing with shorter periods. If you want to go out for longer , over mealtimes,I would really make sure your Mum has been with you, seen your routines and learned how to give the bolus for meals... don't just show her how to do it, you learn much better by actually doing it .Leave written instructions as a back up
Hypo treatments should be no different to when on MDI, make sure that hypo treatments are easy to find, if you have a glucagon pen show it to her and let her read the instructions(there are also a couple of good videos on youtube.) Again leave step by step instructions.
I would again leave instructions on what to do with high levels, the first one would be to phone and contact you, so that you can discuss what to do, so you can (at least at first) make the decision on whether to correct.
Take things gradually, you are only just learning what to do with the pump and even though you deal with it everyday will probably make mistakes or find things don't go the way you expect. You'll learn from your experiences. It will be the same for your Mum so you have to support her as much as you can, otherwise you might find her less available to babysit!