I used to work in a vets and have an animal science degree so I can give you my professional opinion. This will be a very long answer, hopefully it will be useful to anyone else struggling with fleas.
Fleas are particularly bad this year, they are dormant in winter which is why you won't have had an issue. My indoor ragdoll has got them and he doesn't come into contact with other animals at all. Fleas absolutely love all of this warm weather we have been having. They can enter the house through a crack in the window, they can attach to your clothes and you can bring them in that way.
As a general rule OTC flea treatments are ineffective, fleas over the years have become immune to Frontline, Advantage which are both OTC products. Advantage has had zero effect on my Ragdoll (I discovered the fleas when vets were shut and it was all I could get and it did nothing!) Please do not use Bob Martin products!! There have been multiple reported cases of Bob Martin products causing serious harm to pets. I cannot understand why it is still being sold.
If you already have fleas you will see lots of little black bits in your cats fur as well as some actual fleas crawling around. As you already have an infestation you will need to use prescription strength products. The 2 that my vets recommended to me is Stronghold Plus and Bravecto. Stronghold Plus is a monthly Spot On and takes around 24 hours to take affect. Bravecto starts to work within 12 hours and provides protection for 3 months. Prescription flea treatment are significantly cheaper from online pharmacies such as VioVet or Animed Direct. You will need to obtain a prescription from your vet which you will be charged for and will need to upload it to the website you are buying from. This is what I have done and it allowed me to buy a 12 months supply at a reduced cost.
It is also important to note that if you have not been taking your pet to the vet annually they will not issue a prescription without a consultation. The reason being, they are prescribing medication. If any adverse reactions were to happen and they have not seen your pet the vet is liable and could get into serious trouble.
Wash bedding, clothes, any areas your pets frequent with HOT soapy water this kills fleas. Hoover daily and thoroughly. I also recommend treating any rugs and carpets with a product like Virbac which kills all the fleas and eggs within 2 months and protects your carpets for 12 months. Buy a flea comb and comb them regularly.
Flea repellent shampoos aren't really affective unless they are medicated and even then only prescribed in extreme cases most that you can buy OTC contain ingredients that can be toxic such as Kernel Oil, Geranoil, Lavendar.
Please be really careful with natural remedies. ALOT are toxic to cats and most are ineffective. I have seen someone suggest that they use fairy liquid as a shampoo - this is an allergic reaction waiting to happen. There is a lot of really bad if not dangerous advice on the internet around animal care.
Nothing will be as effective as prescription items, pay a prescription fee and get it online. You can still get good products at affordable prices that way.
My vet charges £50.35 for a single pipette of Bravecto which lasts for 3 months. I paid £20 to get a prescription e-mailed to me. I was then able to purchase a 9 month supply at £23.50(ish) per pipette. All in all it cost me around £90 instead of £150 if I bought it from the vets. The prescription is valid for 12 months so I will be placing another order for next year before it expires.