Glad you're going back, you realllllly need a proper asthma review, and regular follow ups. Clearly whatever they prescribed isn't working for you. Make sure they complete an asthma action plan for you. This should contain how much of each inhaler is your normal dose, your lung peak flow rate, and at what frequency/values you should seek medical advice. They should have prescribed you a peak flow meter so you can keep an eye on it yourself (you can buy them for less than the prescription price).
If you only have a brown preventer (beclometasone), as I understand it taking more a day doesn't actually help instantly. It takes a few weeks to build up. Ask them to explain properly how the medication works.
For the hay fever/allergen activated asthma, you could add an antihistamine. Loratadine and cetirizine are relatively cheap (£1 for 30 pills), fexofenadine a bit more (£6 for 30). Do check compatibility with your existing medication.
Finally, an annual Prescription Prepayment Certificate costs £114.50 for 12 months, so that would only be £9.54 a month and be valid for any prescriptions. That's presuming you've checked your eligible for free prescriptions.
Hope you feel better soon, asthma is shit in the cold, and I too hate that it doesn't count as a condition with free prescriptions. I don't need all of them to be free, just the asthma meds.