You will need to get yourself a job before you even think of coming over - you will find it very difficult to gain employment to be honest - unless you have any special skills or experience that would put you ahead of all the other people applying?
The rate of £13,000 is (I think) a full year equivalent, but TAs only work for the school terms which is around 39 weeks, so the actual rate becomes £740 net per month (after taxes) - this is pretty much what a TA friend of mine earns (they actually earn slightly less).
Now, this amount of money is almost certainly not enough to cover your outgoings. Renting a small flat is not going to be possible but you may be able to rent a single room in someones' house - a room in a shared house rents for around £300. It may be difficult to find people willing to rent a room to mother and son. Then you will need to travel to work, support yourself and feed the two of you. You will also need to pay for childcare if you need it. You may be entitled to benefits but I am not sure how much and how quickly you would get them.
Bursaries are not normally given to children under 7 years of age, and they are then usually awarded to children with exceptional talent. Saying that, state schools are free and your child is of an age where they will quickly adapt to a new language so that shouldn't be a problem.
Of course you will not be able to work while you do a PGCE, because these are normally full time courses. There are some courses that allow you to teach whilst you qualify but I expect competition is hard for these places.
In all honesty, I don't think your ideas will work. There are many many people here in the UK who would love to do TA work, but cannot because very few vacancies, lots of competition and very low pay. You will need to check carefully about access to benefits/likelihood of getting a job, etc. If you do get a job you need to accept that you will have to live in a less expensive area (West London is impossible on a salary of even £30,000). You may have to live in assisted accommodation in a high unemployment area. I am sorry, but that is the reality of a household income of £9,500 with young children.