Yes, there is a risk that the boys will end up at different schools, at least temporarily.
Your best option, in my view, is to take up the place for your oldest son. Then, as you've been advised, your youngest son will go somewhere near the top of the waiting list as a sibling.
Do you know what the admission number is for the new school? If it is a multiple of 30, then any appeal which you submit will be treated as an "infant class size" appeal, which means (basically) that you have to demonstrate that the LEA made an error or did not follow proper procedures in refusing a place for your son. (If, though, each class in reception has fewer than 30 children you have a bit more scope for arguing that your son could be admitted without breaching the legal class size limit of 30).
From the sound of it, though, there has been no error and usual procedures have been followed. It's usual to give people a second chance to reply to the offer of a school place. The School Admissions Code 2010 explains the very limited circumstances in which the offer of a place can be withdrawn:
Withdrawing offers of places
1.50 Once an offer of a school place has been made it is only reasonable for an admission authority to withdraw that offer in very limited circumstances. These may include when a parent has failed to respond to an offer within a reasonable time or the admission authority offered the place on the basis of a fraudulent or intentionally misleading application from a parent (for example, a false claim to residence in a catchment area) which effectively denied a place to another child; or where a place was offered under co-ordination by the local authority, not the admission authority, in error. If a parent has not responded to the offer of a place within a reasonable time, the admission authority must remind the parent of the need to respond within a further seven days and point out that the place may be withdrawn if they do not.
If the basis of your appeal is going to be that the LEA should have taken a place away from another child to give it to your son, you will probably not succeed - unless, that is, you can demonstrate that the LEA has not followed its own procedures, but it seems doubtful that any LEA would have procedures in which offers of places were withdrawn on a massive scale.
In any appeal, you need to present positive reasons why your son should be admitted to the school. Presumably your younger son has been offered a place at another school. You need to use the appeal hearing to highlight any compelling reasons why you cannot manage to get both boys to different schools or (indeed) why they could not both go to the school which has a place for your younger son.