Reaction between 2 hydrogen halides
The reaction you are proposing is very unlikely to happen under standard conditions. Taken average bond energies from UC Davis ChemWiki,[1] the approximated heat of reaction would be $\Delta_\mathrm{R}H = 307~\mathrm{kJ\,mol^{-1}}$.
$$\ce{HCl + HF &lt;=&gt; H2 + FCl}$$

The reaction mechanism you are proposing is also not quite feasible. If we would ignore the energetics of this reaction, and we speak hypothetical about this reaction, we would still have to consider an activation step. What we also need to consider is the high affinity of fluorine towards electrons and therefore have to exclude anything that would enhance the heterolytic bond cleavage of hydrogen fluoride. Hence this reaction can only take place in gas phase in an inert atmosphere.
A feasible starting point would be the thermolytical or photolytical cleavage of one of the bonds.
$$\begin{align}
\ce{HCl &amp;-&gt;[\Delta T/ h\nu] H + Cl}\\
\ce{HF  &amp;-&gt;[\Delta T/ h\nu] H + F} \\
\end{align}$$

Then we would need to consider propagation of the reaction. 
$$\begin{align}
\ce{HCl + H/Cl/F &amp;-&gt;{} [ HCl.(H/Cl/F)]^{\ddagger} &amp;&amp;
-&gt; H(H/Cl/F)/Cl(H/Cl/F) + H/Cl}\\
\ce{HF + H/Cl/F &amp;-&gt;{} [ HF.(H/Cl/F)]^{\ddagger} &amp;&amp;
-&gt; H(H/Cl/F)/F(H/Cl/F) + H/F}\\
\end{align}$$
And then finally recombination of the remaining free radicals.

We are basically looking at the reverse mechanism of the reaction of hydrogen and chlorine.[2]

The practical approach is given in orthocresol's answer. Interhalgen compounds, i.e. halogen halides, are formed by reacting the pure halogen gasses. For the reaction of chlorine and fluorine we find the approximate enthalpy of reaction to be $\Delta_\mathrm{R}H = -113~\mathrm{kJ\,mol^{-1}}$.
$$\ce{F2 + Cl2 &lt;=&gt; 2FCl}$$
Side products also include $\ce{ClF3}$ and $\ce{ClF5}$. The reaction itself is exothermic and ignoring entropy should proceed spontaneously.
$$\begin{align}
\ce{Cl2 &amp;-&gt;[h\nu] 2Cl}    &amp;&amp;\text{(start)}\\
\ce{Cl + F2 &amp;-&gt; ClF + F}  &amp;&amp;\text{(propagation)}\\
\ce{F + Cl2 &amp;-&gt; ClF + Cl}\\
\ce{F + Cl &amp;-&gt; ClF}       &amp;&amp;\text{(stop)}\\
\end{align}$$

All reactants are highly reactive and may decompose explosively when other reactants are involved. Even in diluted systems these compounds react with almot everything, see also Why does free chlorine in the stratosphere lose its ozone-depleting potential after about 100,000 reactions?

Conclusion

The formation of interhalogen compounds via their hydrogen halides is energetically not feasible. They can be synthesised from their pure compounds under moderate conditions.
The mechanisms for these reactions can only develop in a closed system as radical chains, which need an initiation step.

Appendix


Average bond energies (Bond dissociation energies; BDE) from UC Davis ChemWiki:
$$\displaystyle\begin{array}{lr}
&amp;\text{BDE}/\mathrm{[kJ\,mol^{-1}]}\\\hline
\ce{HCl} &amp; 427\\
\ce{HF}  &amp; 565\\
\ce{H2}  &amp; 432\\
\ce{FCl} &amp; 253\\\hline
\ce{F2}  &amp; 154\\
\ce{Cl2} &amp; 239\\\hline
\end{array}$$
A. L. Marshall, J. Phys. Chem. 1925, 29 (7), 842–852.
Carl M. Furgason and John W. Moore, J. Chem. Educ. 1943, 20 (1), 41.