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Via mixed integer linear programming with a binary decision variable for each of the $\binom{11}{1}+\binom{11}{3}=176$ possible teams, I have found that the minimum is

15 minutes and 20 seconds,

attainable via teams with times

{10,16,19}, average 15

{11,17,18}, average 15 + 1/3

{12,14,20}, average 15 + 1/3

{13}, average 13

{15}, average 15

Here's an alternative optimal solution, with only one "slow" team:

{10,17,18}, average 15

{11,16,19}, average 15 + 1/3

{12,13,20}, average 15

{14}, average 14

{15}, average 15

Here's the formulation I used. For each team $T$ (subset of cardinality $1$ or $3$), let binary decision variable $x_T$ indicate whether that team is used. Let $c_T$ be the average time for team $T$. Let decision variable $z$ represent $\max_T c_T x_T$. The problem is to minimize $z$ subject to linear constraints \begin{align} \sum_{T: c \in T} x_T &= 1 &&\text{for all cyclists $c$} \tag1\label1\\ c_T x_T &\le z &&\text{for all teams $T$} \tag2\label2 \end{align} Constraint \eqref{1} assigns each cyclist to exactly one team. Constraint \eqref{2} enforces the minimax objective.

Via mixed integer linear programming with a binary decision variable for each of the $\binom{11}{1}+\binom{11}{3}=176$ possible teams, I have found that the minimum is

15 minutes and 20 seconds,

attainable via teams with times

 {10,16,19}, average 15
 {11,17,18}, average 15 + 1/3
 {12,14,20}, average 15 + 1/3
 {13}, average 13
 {15}, average 15

Here's an alternative optimal solution, with only one "slow" team:

 {10,17,18}, average 15
 {11,16,19}, average 15 + 1/3
 {12,13,20}, average 15
 {14}, average 14
 {15}, average 15

Here's the formulation I used. For each team $T$ (subset of cardinality $1$ or $3$), let $a_T$ be the average time and let binary decision variable $x_T$ indicate whether that team is used. Let decision variable $z$ represent $\max_T a_T x_T$. The problem is to minimize $z$ subject to linear constraints \begin{align} \sum_{T: c \in T} x_T &= 1 &&\text{for all cyclists $c$} \tag1\label1\\ a_T x_T &\le z &&\text{for all teams $T$} \tag2\label2 \end{align} Constraint \eqref{1} assigns each cyclist to exactly one team. Constraint \eqref{2} enforces the minimax objective.

Via mixed integer linear programming with a binary decision variable for each of the $\binom{11}{1}+\binom{11}{3}=176$ possible teams, I have found that the minimum is

15 minutes and 20 seconds,

attainable via teams with times

{10,16,19}, average 15

{11,17,18}, average 15 + 1/3

{12,14,20}, average 15 + 1/3

{13}, average 13

{15}, average 15

Here's an alternative optimal solution, with only one "slow" team:

{10,17,18}, average 15

{11,16,19}, average 15 + 1/3

{12,13,20}, average 15

{14}, average 14

{15}, average 15

Via mixed integer linear programming with a binary decision variable for each of the $\binom{11}{1}+\binom{11}{3}=176$ possible teams, I have found that the minimum is

15 minutes and 20 seconds,

attainable via teams with times

{10,16,19}, average 15

{11,17,18}, average 15 + 1/3

{12,14,20}, average 15 + 1/3

{13}, average 13

{15}, average 15

Here's an alternative optimal solution, with only one "slow" team:

{10,17,18}, average 15

{11,16,19}, average 15 + 1/3

{12,13,20}, average 15

{14}, average 14

{15}, average 15

Here's the formulation I used. For each team $T$ (subset of cardinality $1$ or $3$), let binary decision variable $x_T$ indicate whether that team is used. Let $c_T$ be the average time for team $T$. Let decision variable $z$ represent $\max_T c_T x_T$. The problem is to minimize $z$ subject to linear constraints \begin{align} \sum_{T: c \in T} x_T &= 1 &&\text{for all cyclists $c$} \tag1\label1\\ c_T x_T &\le z &&\text{for all teams $T$} \tag2\label2 \end{align} Constraint \eqref{1} assigns each cyclist to exactly one team. Constraint \eqref{2} enforces the minimax objective.

Via mixed integer linear programming, I have found that the minimum is

15 minutes and 20 seconds,

attainable via teams with times

{10,16,19}, average 15

{11,17,18}, average 15 + 1/3

{12,14,20}, average 15 + 1/3

{13}, average 13

{15}, average 15

Here's an alternative optimal solution, with only one "slow" team:

{10,17,18}, average 15

{11,16,19}, average 15 + 1/3

{12,13,20}, average 15

{14}, average 14

{15}, average 15

Via mixed integer linear programming with a binary decision variable for each of the $\binom{11}{1}+\binom{11}{3}=176$ possible teams, I have found that the minimum is

15 minutes and 20 seconds,

attainable via teams with times

{10,16,19}, average 15

{11,17,18}, average 15 + 1/3

{12,14,20}, average 15 + 1/3

{13}, average 13

{15}, average 15

Here's an alternative optimal solution, with only one "slow" team:

{10,17,18}, average 15

{11,16,19}, average 15 + 1/3

{12,13,20}, average 15

{14}, average 14

{15}, average 15