Well I must write a warning before I offer advice - my DP was away this week and I tried to do bath/bed by myself for the first time EVER. Even though DD1 is now 2.6 and DD2 is nearly 15months, I just couldn't cope and won't be doing it again if I can help it.
So, having established that I am clueless, I would say this. I have a small age gap too. In the early days, I did just skip bath sometimes with DD1. You can do a thorough wash of hands/face, brush teeth and make that do a lot of time if your standards are as low as mine.
Are you by yourself? Is there anyone who can help you (DH, DP, family)? In the early weeks, take any help you can. Someone holding/jiggling the baby while you quickly do bath/bed with your eldest would be ideal. I imagine yiou don't have that option though?
Timing- can you do anything about timing? Remember things change quickly at this stage with a little baby. What is cluster feeding and clinginess might in a week or two be different. So it's just survive however you can.
Could DD1 go to bed earlier/later so that it doesn't coincide exactly with cluster feeding or unsettled crying for DD2?
If you can get DD2 to even have a little nap (even if she's not ready for bed til much later in the evening) in a rocking cradle, bouncy chair, whatever, then you could whizz through bedtime. Bouncy chair in the bathroom and bedroom?
However, ime, it was unavoidable - I had to sometimes let DD2 cry. I couldn't see another way round it. Upsetting but necessary really. I owed it to DD1 (in my mind) to give her 10 mins of bedtime without DD2's constant presence. I would console myself by thinking that DD2 would get all my attention once DD1 was in bed (and all night, but that's another story )
Hope you figure something out but importantly, you don't need to feel you have a "routine" all sorted out right now. Just do what it takes to survive and worry about the routine when DD2 is a bit older.
(Or like me, just try to ensure that you NEVER have to do bed/bath by yourself again for at least 15 months )