Wolfram Language (Mathematica), 100 bytes

Nest[#+1//.i_/;!PrimeQ@FromDigits[##&@@#~Table~#&/@(l=IntegerDigits@i)]||Min@l<1:>NextPrime@i&,1,#]&

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Jonathan Frech saved 3 bytes

and -7 bytes from JungHwan Min

-15 bytes from Martin Ender

also thanks to Jenny Mathy

Wolfram Language (Mathematica), 100 bytes

Nest[#+1//.i_/;!PrimeQ@FromDigits[##&@@#~Table~#&/@(l=IntegerDigits@i)]||Min@l<1:>NextPrime@i&,1,#]&

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Jonathan Frech saved 3 bytes

and -7 bytes from JungHwan Min

-15 bytes from Martin Ender

also thanks to Jenny Mathy

[Wolfram Language (Mathematica)][1], 115 bytes

(For[i=t=1,t<=#,PrimeQ@FromDigits[Join@@(#~Table~#&/@(l=IntegerDigits)@(p=Prime)@i++)]&&l@p@i~FreeQ~0&&++t];p@--i)&

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Jonathan Frech saved 3 bytes

and -7 bytes from JungHwan Min [1]: http://%20https://www.wolfram.com/wolframscript/

Wolfram Language (Mathematica), 100 bytes

Nest[#+1//.i_/;!PrimeQ@FromDigits[##&@@#~Table~#&/@(l=IntegerDigits@i)]||Min@l<1:>NextPrime@i&,1,#]&

Try it online!

Jonathan Frech saved 3 bytes

and -7 bytes from JungHwan Min

-15 bytes from Martin Ender

also thanks to Jenny Mathy

[Wolfram Language (Mathematica)][1], 115 bytes

(For[i=t=1,t<=#,PrimeQ@FromDigits[Join@@(#~Table~#&/@(l=IntegerDigits)@(p=Prime)@i++)]&&l@p@i~FreeQ~0&&++t];p@--i)&

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Jonathan Frech saved 3 bytes

and -7 bytes from jungminhwan [1]: http://%20https://www.wolfram.com/wolframscript/

[Wolfram Language (Mathematica)][1], 115 bytes

(For[i=t=1,t<=#,PrimeQ@FromDigits[Join@@(#~Table~#&/@(l=IntegerDigits)@(p=Prime)@i++)]&&l@p@i~FreeQ~0&&++t];p@--i)&

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Jonathan Frech saved 3 bytes

and -7 bytes from JungHwan Min [1]: http://%20https://www.wolfram.com/wolframscript/