Worries about weight

Hiit sessions don't necessarily have to include press ups/ push ups. Hiit can be done with just about any exercise even running. To introduce Hiit into your running every so often run flat out the maximum speed you can go for about 30seconds then resume normal pace for a whilst rinse and repeat.
You say you run 20k+ a week over how many running sessions is that and is it all done on the treadmill or actually cross country /road running ?
Gaining fat is most likely the result of eating too much or perhaps eating too much refined sugars or white breads/pasta/rice etc. If this is the case either try to eat slightly less or switch to wholegrain alternatives. It is also the likely outcome of your weight gain too although that could be down to medical issues like time of the month etc.
Lastly if you absolutely want to be able to do pushups then you need to strengthen your arms, back and chest - a good starting point would be to get into a plank position and hold it for as long as you can rinse and repeat several times. Or/and do weight lifting movements like just research the form for different exercises like bi-cep curls and do those without any weights - do them slowly and get your breathing patterns right you should feel your arms resisting the movements. Gradually as the movements get easier use weights - start off light with tins of food from the cupboard as an example, before moving onto actual weights.

The ones in the classes at the gym are quite strength heavy and I’m not really motivated enough and don’t have the space to do it at home.
Over about 3 sessions, sometimes I do 5 runs a week and each run is at least 5k and I try to do 12k or over every Sunday, though this was slowly increase as I’m marathon training. It’s a mixture of treadmill and outside, though on treadmill I put it on .5 incline as apparently this imitates the road better.

I definitely could cut out carbs like bread pasta and rice, does wholemeal make a noteable difference or is it better to cut out all together? Perhaps I could switch it to a pulse alternative would that be better?
I’m early 20s and I’m worrying my metabolism has just slowed down a lot.

I definitely could plank more. Thanks for the tips RE bicep curls that’s a very good idea

Wholemeal can help!

It s a big jump in weight for someone that exercises a fair bit. Maybe if you are really worried go to your GP - check for hypothyroidism

Do they test this via simple blood test? Would make sense, I sit with a hot water bottle freezing every day at work!

Yeah you just test for a hormone called TSH. Do go check it out! Also feel free to ask me any questions

Decided I probably don’t have it and need to stop making excuses for getting large haha. I definitely could do with cutting out carbs. Would it be mad to go on a keto diet and marathon train?

I think I could manage a calorie deficit of 500 a day (my BMR is rubbish at 1300 and I have a desk job), really failing to see how people do deficits of 1000!