JAJ, you are of course quite right about home schooling. But it does of course need to be the case that the parents personally have the necessary skills, or buy in those skills at huge expense.
jpfranco, if you are looking for a job in Birmingham, and you succeed in that endeavour, don't hesitate to contact me to see if I can help about school selection. Unfortunately the standard of schools in Birmingham, and no doubt the rest of the UK, does vary considerably and at first glance it is not always obvious which are good and which are bad. Indeed if you find a job before you start renting somewhere permanent, get in touch because you might want to select residential accommodation near to good schools .... and away from bad schools.
Your daughter who is 12 next April will be in Year 7 in the school year that starts in September .... and thus will be in secondary school. Your son, aged 9 next April, will be in Year 4 starting in September, and thus will be in Primary School. So it is not a question of you finding one school for your two children .... you will need to find two separate schools, one Secondary and one Primary.
do you think if they arrive around Christmas into the UK, they could start clases in January without a problem?
Obviously the sooner you know where you will be living, and thus which schools are around, the better chance you have of ensuring they have schools to go to early in January. If you leave it to the last moment, near the end of the school term just before Christmas, you might not have completed arrangements before the start of the new term in January.
Particularly as regards your son, now aged 8, I probably don't need to tell you that children pick up language so much quicker than adults. My step-daughter was 6 when she moved from Thailand to here in Birmingham. Now she is 12 and it is impossible to detect that she has not lived here all her life, well apart from when she is chatting to her mother in Thai! So your children's English will improve rapidly!
