You will need to check dates very carefully.
You COULD lose your primary place (actually, that is really quite likely) and you may also fail to get a secondary place because you are not resident at your address on the relevant dates (this would depend on the precise LEA / school you are applyng to, how over-subscribed they are and how strictly they check).
I don't think that there is anything wrong with taking your child out of school to travel and home educate per se. I am an ex HEer myself, after all. However, you can't expect to 'turn off' the school system and then 'turn it back on' exactly as you choose.
If you are happy to HE your children for a while after they come back OR have them transfer to another primary school with places on your return AND you have checked carefully with the LA that being 'not resident' over that precise period won't impact on your secondary application, then go for it - it sounds amazing. However, youy have to accept the downside risks, and regard the upside benefits as being sufficiently large.
For those from a Traveller background, educational outcomes are statistically SO poor (the lowest performing of all ethnic groups, historically) that specific arrangements for ensuring that they can continue to access schools are necessary. Some travellers are dual registered (so a child whose family typically travels between 2 different sites over the course of a year is allowed to be registered at 2 schools), and specific arrangements can be invoked if a traveller child turns up to reguister at a full school. However, in general, Travelleras don't access 'high profile outstanding city / suburban schools' - they are more likely to access rural schools, often in relatively less affluent areas, simply because that is where both formal and informal sites tend to be situated.