I am going to throw in a couple of more things to bear in mind.
Consider going to boots and buy some anti dustmite bedding.
Dont warm any clothes on a radiator before putting them on your baby.
Boil wash all bedding.
These steps alone helped my son, when he was younger. Dustmite allergy is very common.
The cough you mention may be linked to this, ie rhinitus, and if you remove a the high rate of dustmites in his cot/bedroom and this improves, well, its worth a try!!
GP's are jack of all trades, but are often not expert in many conditions.
As for any allergy related help in the NHS, its all pityful, patchy and sometimes a GP is wrong.
It's not always their fault, they cant help it, bless them!!
Find a recently qualifed GP, they will be more knowledgeble about allergic conditions.
Carry one researching all you can about ezcema.
If your child shows any of these other signs, poor to slow weigh gain,
and a terrible sleep pattern,
as well as ezcema, take care when introducing soilds, as all these are pointers, esp for a breastfed baby under three months, to food allergy.
Asthma in infants is difficult to diagnose, and often docs will wait until a year to do this.
You are right to keep the baby away from ciggaretes.
My son didnt sleep for five soild hours until he was 9 months old. His sleep pattern( HA!) was two or three seperate hours a night, and 20 min cat nap during the day.
His ezcema started at day 10 after birth.
His weight gain was very slow to poor, and even now at 10 ( this friday) he is still underweight.
He was fully breastfed for 18 months and this worked well, his older brothers were completely opposite to our allergic boy. Top of the charts for height and weight, and health wise robust in the extreme.
My second son was so big that the health visitors kept asking if I was over feeding him on formula , but none of my children ever had formula!! ( but thats a different story).
But I have been where you are now, and know how it feels.