Hey benjo,
Couple more suggestions from me:
Even though you're not saving much at the moment, you seem to be saving some, and paying off your mum, so I agree that not giving notice until you have this sorted is the right thing to do. £700 isn't that bad for London, I imagine you're looking at £400-500 sharing, plus bills, but is that still close to your office? Make sure you factor in any additional travel costs, cos you'll assume you'll be saving 200 quid a month but with extra costs of travel or moving it might be more like £50. Factor in cost of van hire etc if you gonna need it, and make sure you have not only first month's rent + deposit + admin but some contingency as well. Also, if you pay off your mum then it won't be as awkward to ask for a couple of hundred quid again when you do move.
With regard to your work, they might be willing to give you a significant advance on earnings, but companies often do loans for certain things, like long-term train tickets, or bike schemes, check those out. They usually have a tax break, so you're not just paying off the loan at 0% but you save on tax a bit too, might only save you a bit of cash.
Next, start applying for 0% credit cards, moneysupermarket or moneysavingexpert will have lists of the 0% transfer rates etc, some have discounted rates, some have 0 rate for transferring other balances across, start looking around (perhaps after you've checked your credit so you don't waste time applying for things you won't get). Maybe do a search for cards that a more generous on credit checks as well, if you successfully get one and use it for a month or two then that will help you chance of getting other, but also it would mean that you could take out money on it and then if you get a 0% transfer credit card you can move the whole balance across and pay no interest for a year or so!
Also, I'd suggest you speak to your local council where you live, find out about any local loans or schemes or grants. They can also double check there's no in work benefits you could be applying for. I doubt there will be, but doesn't hurt to ask. Which borough do you live in? If not council then Citizen's Advice Bureau.
With regard to income, you could consider getting a weekend job, like bar work or something like that, just go get you over this hurdle. If you make another £40 a week then that's another £170 a month you could be saving. £350 in two months, plus the £700 a month you already pay, plus you have £400. so that's £1,450. It might also help with credit card applications. Based on the fact you're already saving even £25 a week would help a lot, done consistently. If you don't want to get another job, then consider selling stuff on ebay or that sort of thing. Alternatively/as well, look to reduce costs, if you save £20 a week on some form of expenditure, that saves you getting a job and is almost an extra £100 a month.
Lastly, where you move to is important. Given your situation, moving to somewhere where you are living with the owner of the flat/house is likely to mean you could get away with less deposit-wise, as they will be there to keep an eye on you, so they'll be a bit less worried about you. Or, since you don't HAVE to move, shop around for a similiar deal as you have now (I've no idea how you managed it, but if one place let you move in without deposit, there must be another cheaper one elsewhere). There are also some places where you can live rent free as a sort of caretaker, guardians I think they're called, in London demand is very high, and sometimes the buildings are a bit weird, but you could register on their site regardless, it doesn't hurt, even if it takes a couple of years to get an offer of a place.
My final note, don't be too keen to move somewhere cheaper with a deposit, landlords often find some sort of excuse to keep a sizeable chunk of your deposit at the end of the tenancy, there's ways to mitigate it, but it's always a risk. So you might think you'll be saving £100 a month, but if you stay there a year and end up losing £600 of your deposit then you're only really £50 a month up (for example).
both about money and women.
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