Hi there. Sounds like you're working really hard at these naps!
To be honest she sounds UT to me. 30 mins is an incredibly short A time and the resistance to nap (crying) and being hard to settle sounds more like she just knows she doesn't want to go to sleep yet.
Mine was also 2 weeks over and I found his A times and other developmental areas were more like his due date rather than his birth date, so at 10 weeks he was prob on a routine more suited to a 12 week old where as you have your LO on a routine with shorter A time than a new born. You may not think a 10 week old can know her own needs but at this age mine made huge efforts to communicate his needs to me, took me a while to figure out what he was saying! I'd planned all naps in the travel cot/play pen in the family room up to 6 months (following SIDS guides to have LO sleep in the room I was in) but at 10 weeks he suddenly refused to nap (he was already an independent sleeper, I did WD and put him down awake), instead of going to sleep he just screamed his head off. After about 2 days of this in desperation I took him to his night bed (in my bedroom) where he instantly quieted down, I put him in his bed and he smiled and went instantly to sleep! I was pretty shocked I can tell you!
how long should I let her cry for in the cot
Please don't let her cry. At this age she will burn too many calories which will just make her hungry.
pat shh doesn't really work with her and picking her up stops the crying but just wakes her up more, she just starts pushing off your chest and looking around at anything she can
This sounds like typical UT behaviour. IMO she is crying to communicate that she needs you to get her up, and once she is up in your arms she is happy and interested in the world. Shush/pat will not sooth an UT baby to sleep, more likely it will frustrate her because she knows you are trying to get her to sleep when she doesn't want that.
would you say a good bed time routine is more or less important than getting her to sleep quicker when OT.. ie. if shes missed her nap and already wired should we just keep trying to get her to sleep or stop and bath, massage etc at bedtime even though it means taking her to a different room and doing more stimulating activities?
Hopefully you can avoid all this by extending her other A times to a reasonable length, spreading her naps across the day and therefore avoiding the 3hr A time before BT. However, to answer your question, with a super long A time and a super OT baby I would skip anything that can be skipped in the BT routine and just get her fed, clean nappy and in bed. Missing a bath and massage isn't going to ruin your BT routine. Mine hated baths at that age so I topped and tailed, no way could he handle a bath or massage before bed. Maybe move bath time to another time of day if she is repeatedly OT at BT.
You might not want to hear this but I tend to agree with your HV about getting out. It hard work (at best) and can be soul destroying staying home all day, especially if the majority of that time is spent in a darkened room trying to get your baby to sleep. I really do think it does everyone good to get out, get sun, get air. And tbh if you want to ever leave the house then starting some naps in the stroller/sling now are going to be pretty handy later on. Having a LO who won't even CN out and about can feel very restricting indeed. I found mine slept better when he'd seen some of the world too, babies are such inquisitive little things, eager to learn everything about the world and a certain level of stimulation is needed to help them feel mentally tired enough to sleep.
Many hugs. I know you have your baby's best interests at the centre of all your plans, being concerned about going out or letting her get OT is admirable whilst can also cause some difficulties.
I would increase her A time directly to a minimum of 1hr and over the next few days increase by another 15 mins, hold for 3 days then re-evaluate. She may need an A time of 1hr 20 to 1hr 30 if she sleeps more like a 12 wk old. In any event going too slowly to get those A times up won't be particularly helpful, it perpetuates the problems of UT, fighting naps, short naps and long A times to bed. She need to be tired for each and every nap to reach BT without becoming OT.
hope this helps x